Worked problem

Differential-Pressure Flow: Why 50% Signal Is Not 50% Flow

Convert a 12 mA signal from a linear differential-pressure transmitter into flow with the square-root law, and compare a flow-linear output.

Controls & Instrumentation · Last reviewed

Given

An orifice plate measures a flow that is 0 to 500 gpm. The differential-pressure (DP) transmitter is ranged 0 to 100 inH₂O for 4 to 20 mA. The flow through an orifice varies with the square root of the DP, so 100 inH₂O corresponds to 500 gpm.

Find

The flow when the signal is 12 mA, the signal at 50% flow, and the result when the transmitter itself extracts the square root.

Solution

Q = Q(max) × √(DP ÷ DP(max))
  1. Signal 12 mA is the middle of the 4 to 20 mA span, so the DP is 50% of range: 50 inH₂O.
  2. Flow = 500 × √0.50 = 500 × 0.7071 = 353.6 gpm, which is 70.7% of full scale.
  3. For 50% flow (250 gpm): DP = 100 × 0.50² = 25 inH₂O, 25% of range, so the signal is 4 + 0.25 × 16 = 8 mA.
  4. If the transmitter extracts the square root, its output is linear in flow, and 12 mA means 50% of flow = 250 gpm.

Answer

linear-DP signal of 12 mA = 353.6 gpm (70.7%) | 50% flow = 8 mA | flow-linear transmitter: 12 mA = 250 gpm
DP (% of range)Flow (% of max)Linear-DP signalFlow-linear signal
25%50%8 mA12 mA
50%70.7%12 mA15.3 mA
75%86.6%16 mA17.9 mA
100%100%20 mA20 mA

Where the confusion comes from

  • A linear DP signal is linear in pressure, not in flow. Reading 12 mA as 50% flow gives 250 gpm instead of 353.6 gpm, an error of 29%.
  • The conversion must happen exactly once: either in the transmitter (flow-linear output) or in the controller or PLC (square-root function). Doing it twice, or not at all, gives a wrong flow.
  • At low flow the DP is very small, and the signal noise becomes a large fraction of the flow. Many systems cut off the flow indication below a low-flow limit such as 5 to 10% of range.

Common mistakes

  • Scaling the flow linearly from the 4 to 20 mA signal when the transmitter outputs DP.
  • Applying the square root twice, once in the transmitter and again in the PLC.
  • Forgetting that the totalizer then integrates the wrong flow.
  • Using the same range for different fluids. The DP at full flow depends on the density.

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

More worked problems

Questions

Why is flow proportional to the square root of DP?

The DP across an orifice or venturi comes from the change in velocity head, which scales with the square of the velocity. So velocity, and flow, scales with the square root of DP.

How do I know how my transmitter is configured?

Check the transmitter configuration for the output mode (linear or square root), the range and the units. Confirm in the loop documentation and the PLC scaling.

Does the turndown limit apply?

Yes. A DP flow meter loses accuracy at low flow, so the usable range is typically limited to about 10:1 or less.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

← All worked problems