Orifice Flow Calculator
Estimate liquid flow through a circular sharp-edged orifice from differential pressure, bore diameter, density, discharge coefficient, and expansion factor. Results include US gpm, cfs, bore velocity, beta ratio, ideal flow, mass flow, and differential-pressure energy rate.
Coefficient-supplied model. The calculator does not derive an ISO 5167 or API coefficient from Reynolds number, tap location, pipe roughness, and plate geometry. Use a validated standard or calibration for metering work.
Discharge estimate
Orifice-equation trace
Area = π × (0.083333 ft)² ÷ 4 = 0.005454 ft² Q = 0.6200 × 1.0000 × 0.005454 × √(2 × 32.174 × 1,440.000 ÷ 62.400) Q = 0.130309 ft³/s = 58.487 US gpm
The differential creates an ideal jet; Cd corrects reality
Driving pressure
Differential pressure is upstream minus downstream pressure at defined taps. Flow is proportional to the square root of ΔP, so four times the differential produces twice the modeled flow when density and coefficients remain constant.
Opening area
Circular bore area is πd²/4. Flow therefore responds to diameter squared. Plate bore must be measured under the governing standard because edge damage, deposits, thermal expansion, and manufacturing tolerance can affect the result.
Discharge coefficient
Cd accounts for contraction, velocity distribution, friction, and the relationship between measured differential and actual discharge. The Bureau of Reclamation describes it as an experimentally determined correction; it is not one constant for every orifice.
The expansion factor Y accounts for density change in compressible flow formulations. For this incompressible liquid screen it should generally remain 1.000. Gas, steam, flashing liquid, cavitating restriction, and choked flow require an appropriate compressible-flow standard and thermodynamic properties.
Worked example: 1-inch bore, 10 psi across water
The default bore is one inch, or 0.083333 foot. Its area is approximately 0.005454 square foot. Ten psi equals 1,440 pounds-force per square foot. With 62.4 lb/ft³ density and gc = 32.174, the ideal velocity term is approximately 38.535 ft/s.
Multiplying area by that ideal velocity gives 0.210176 ft³/s before correction, or about 94.33 US gpm. The entered Cd of 0.62 and liquid Y of 1.00 reduce the estimate to 0.130309 ft³/s, or 58.487 US gpm. The modeled mass flow is 487.88 lb/min.
The one-inch bore in a four-inch-ID pipe gives beta ratio β = d/D = 0.250. This calculator displays beta but does not calculate a beta-dependent velocity-of-approach factor or standardized coefficient. Use the equation and coefficient convention required by the selected metering standard without double-counting corrections.
The product of differential pressure and actual volumetric flow is about 254.4 W. It represents the rate associated with the measured pressure differential, not necessarily permanent hydraulic loss. Some pressure recovers downstream; the permanent loss depends on meter geometry, beta, Reynolds number, and installation.
Why differential-flow transmitters use square-root extraction
Because Q is proportional to √ΔP, the raw differential signal is quadratic with flow. At one-half flow, differential is one-quarter. At one-tenth flow, differential is one-hundredth. A transmitter, control system, or flow computer may perform square-root extraction so the reported signal is linear in flow.
The default scenario illustrates the relation: 2.5 psi produces 29.24 gpm, 10 psi produces 58.49 gpm, and 40 psi produces 116.97 gpm under fixed properties and coefficients. Each fourfold differential step doubles flow.
Low-flow accuracy can degrade because differential becomes small relative to transmitter zero, noise, impulse-line head, calibration uncertainty, and process fluctuations. Do not extend a meter below its validated turndown merely because the square-root equation returns a number. Bidirectional or pulsating flow needs additional treatment.
Cd depends on geometry, Reynolds number, and installation
Free or submerged opening
A sharp-edged opening in a tank or wall forms a contracted jet. Reclamation’s Water Measurement Manual notes a coefficient near 0.61 for certain fully contracted conditions and emphasizes that boundaries, approach velocity, edge geometry, and calibration influence the effective value.
Orifice plate in a pipe
Pressure-conduit meters use standardized plate geometry, beta range, tap locations, straight-run conditions, Reynolds corrections, expansibility, and precise dimensions. Reclamation warns that misalignment and disturbed velocity distribution can materially affect an orifice plate coefficient.
Measurement details can dominate uncertainty
| Input or condition | Why it matters | Verification action |
|---|---|---|
| Differential pressure | Q varies with its square root; zero error and impulse-line effects matter most at low DP. | Calibrate range and zero, verify wet/dry legs, purge gas or liquid pockets, and correct elevation head. |
| Bore diameter | Area varies with diameter squared and standardized calculations may use thermal expansion. | Use inspected bore, edge condition, material temperature, plate tag, and standard tolerances. |
| Pipe inside diameter | Sets beta and velocity-of-approach effects in pipe-meter methods. | Use actual standardized meter-tube diameter, not nominal pipe size. |
| Density | Flow varies inversely with square root of density; mass flow also multiplies by density. | Use process pressure, temperature, composition, and an authoritative property method. |
| Cd and Y | Both multiply flow directly, so a one-percent coefficient error is a one-percent flow error. | Match standard edition, meter geometry, Reynolds range, tap location, and fluid phase. |
| Flow profile | Swirl, elbows, valves, reducers, fouling, and pulsation can invalidate calibration assumptions. | Meet straight-run and flow-conditioner requirements and inspect upstream/downstream piping. |
A rigorous uncertainty analysis propagates bore, pipe diameter, differential, density, coefficient, expansion factor, temperature, pressure, and installation uncertainties. For billing, environmental reporting, regulated testing, or process safety, use the mandated standard and traceable instrumentation.
Model boundaries
- One-phase liquid: density is treated as constant across the restriction. No gas, steam, flashing, dissolved-gas release, cavitation, slurry slip, or multiphase behavior is modeled.
- User-supplied coefficients: Cd and Y are not derived from beta, Reynolds number, roughness, tap position, plate thickness, edge radius, or approach profile.
- Positive differential: the calculator assumes flow in the stated upstream-to-downstream direction. It does not resolve reverse or oscillatory flow.
- No permanent-loss prediction: ΔP × Q is displayed as a differential energy rate, but pressure recovery and net system loss require the appropriate meter model.
- No uncertainty certificate: rounded outputs are engineering estimates and do not imply measurement accuracy.
A metering verification workflow
- Define the primary element. Record plate, bore, pipe, material, edge, taps, orientation, and standard or calibration basis.
- Define the fluid. Establish phase, composition, pressure, temperature, density, viscosity, and risk of flashing or cavitation.
- Confirm applicability. Check beta, Reynolds number, straight run, flow direction, pulsation, swirl, and standard limits.
- Configure instruments. Set transmitter range, square-root location, zero suppression/elevation, damping, units, alarm, and low-flow cutoff coherently.
- Install impulse lines correctly. Manage slope, condensation, heat tracing, seals, manifolds, leaks, gas pockets, liquid legs, and ambient effects.
- Calibrate and inspect. Maintain traceability for DP, pressure, temperature, density, bore, and flow computer; inspect plate and piping.
- Reconcile mass and energy. Compare flow with tank inventory, pump curve, control-valve behavior, and independent measurements.
Frequently asked questions
Why is Cd below one?
A sharp restriction produces jet contraction and losses, so actual discharge is below the ideal area-times-Bernoulli-velocity result under the selected convention. Cd packages those effects and depends on geometry and operating regime.
Should expansion factor be one for water?
For an ordinary incompressible single-phase water estimate, use Y = 1.000. Compressible gas and steam metering use an expansibility factor from the applicable standard. Flashing and cavitating liquids need different models.
Does doubling pressure differential double flow?
No. Flow follows the square root of differential. Doubling ΔP multiplies flow by √2, about 1.414. Quadrupling ΔP doubles flow when coefficients and density stay fixed.
Can I use nominal pipe size for beta?
No. Use the actual inside diameter defined by the applicable standard at the relevant temperature. Schedule, material, lining, corrosion, deposits, and measurement location make nominal size inadequate.
Is ΔP the permanent pressure loss?
Not generally. The taps measure a differential around the restriction, and some downstream pressure recovers. Permanent loss depends on element type, beta, Reynolds number, and geometry and requires a separate validated relationship.
Can this calculate gas flow?
No. Gas density changes across the restriction and may approach choked conditions. Use a compressible-flow method with upstream absolute pressure, temperature, composition, isentropic behavior, expansibility, and the governing meter standard.
Use the estimated velocity and fluid properties in the Reynolds number flow calculator to check whether the assumed flow regime is reasonable.
References
- U.S. Bureau of Reclamation, Water Measurement Manual: Differential Head Flowmeters
- U.S. Bureau of Reclamation, Water Measurement Manual: Orifice Relationships
- U.S. Bureau of Reclamation, Conditions for Accuracy of Fully Contracted Orifices
- U.S. Bureau of Reclamation, Water Measurement Manual
Educational incompressible estimate only. Verify fluid phase and properties, primary-element geometry, coefficients, Reynolds range, tap and straight-run requirements, calibration, permanent loss, uncertainty, and qualified meter design.