CALCULATED MASS FLOW
—
kg / hr
| Nm³/hr (ref 0°C, 1.01325 bar) | — |
| Sm³/hr (ref 15°C, 1.01325 bar) | — |
| Vel. in Pipe | — |
| Reynolds No. (pipe) | — |
| Expansibility factor Y | — |
| Velocity of Approach E | — |
| Cd — ISO RHG f(Re,β,tap) [NOT constant] | — |
| Perm. Pressure Loss (% of ΔP measured) | — |
| Perm. Loss (absolute) | — |
| Flow Uncertainty ±U (k=2) | — |
| ΔP/P₁ Ratio (ISO limit <0.25 for Y validity) | — |
An orifice plate is a thin, flat disc with a precisely machined circular hole (the bore) installed perpendicular to a pipe's axis. When fluid flows through the pipe, the orifice forces the entire stream through a reduced area — velocity rises sharply and static pressure drops. Measuring that pressure differential tells you the flow rate.
This works because energy is conserved along a streamline. Bernoulli's principle states that for steady, incompressible, inviscid flow, the sum of pressure energy, kinetic energy, and potential energy is constant. When cross-sectional area decreases, continuity (mass conservation) demands velocity increase, and Bernoulli requires that velocity increase to come at the expense of pressure. The result is a measurable pressure differential directly related to flow rate.
Orifice plates are standardised under ISO 5167-2 and AGA Report No. 3 (for natural gas). When installed correctly, they achieve measurement uncertainties of ±0.5% to ±2% without individual calibration — purely by conforming to dimensional tolerances and installation requirements. This remarkable accuracy from a passive device with no moving parts explains why orifice plates remain dominant in industrial metering after more than a century of use.
The volumetric flow rate through an orifice plate per ISO 5167-2 is:
The term 1/√(1−β⁴) is the velocity of approach factor E. It corrects for the fact that fluid already has measurable velocity in the pipe before reaching the orifice. For β = 0.3, E = 1.004 (negligible). For β = 0.7, E = 1.228 — a 23% correction that cannot be ignored.
For mass flow rate, rearrange using ṁ = ρQ. Note that ρ moves inside the square root:
For compressible gases, apply the isentropic expansion factor Y (ε in some standards):
Cd is the most critical parameter. It accounts for real-fluid effects — viscous losses, turbulence, and the jet contraction at the vena contracta — that ideal Bernoulli theory ignores. ISO 5167 uses the Reader-Harris/Gallagher (RG) equation, derived by regression of ~16,000 calibration data points from 9 independent laboratories:
Because Cd depends on Re_D, and Re_D depends on velocity (and hence Q and Cd), the solution is iterative — start with Cd = 0.61, compute Q and Re_D, update Cd, repeat until convergence (typically 3–5 iterations, |ΔCd| < 0.0001). This calculator performs this iteration automatically.
Sharp-edged, standard: 0.60–0.63
Quadrant-edge (low Re service): 0.77–0.82
Conical entrance: 0.73–0.80
Rounded entrance (nozzle-like): 0.95–0.99
Eccentric / segmental bore: 0.60–0.63
The upstream edge must be sharp. A worn edge (radius e) increases Cd and causes over-reading. ISO 5167 specifies maximum edge radius: e/d ≤ 0.0004. For a 100 mm bore, that is 0.04 mm — less than a human hair. Even modest rounding (e/d = 0.01) increases Cd by ~0.7%, causing ~0.7% positive flow error. After years in erosive service, edge wear can cause 3–5% systematic over-reading.
The beta ratio β = d/D is the central design variable. It determines the differential pressure signal at a given flow rate, the permanent pressure loss, the required straight run, and the measurement uncertainty.
| β Range | ΔP Signal | Permanent Loss | Uncertainty | Best For |
|---|---|---|---|---|
| 0.20–0.30 | Very high — may need small-range DP transmitter | ~90% of ΔP | Higher — RG eq. less accurate at extremes | Rarely used; very low flow, small pipes |
| 0.30–0.50 | High | ~75–85% of ΔP | Good (±0.6%) | General process metering |
| 0.50–0.65 | Moderate | ~55–70% of ΔP | Best (±0.5%) | Preferred range — lowest uncertainty per ISO 5167 |
| 0.65–0.75 | Lower — needs wider-range DP transmitter | ~40–55% of ΔP | Good (±0.6%) | High flow, energy recovery important, large pipes |
| Tap Type | Upstream Tap Location | Downstream Tap Location | Region / Standard |
|---|---|---|---|
| Corner Taps | Flush with plate face upstream | Flush with plate face downstream | Europe; ISO 5167; D < 50 mm |
| Flange Taps | 25.4 mm (1 inch) upstream of plate | 25.4 mm (1 inch) downstream | North America; AGA-3; oil & gas |
| D and D/2 Taps | 1D upstream of plate | 0.5D downstream | Large pipes; near vena contracta position |
Minimum upstream straight pipe (from nearest fitting to orifice plate) per ISO 5167, for β = 0.6:
| Upstream Disturbance | Required Straight Run (× D) |
|---|---|
| Single 90° elbow | 18D |
| Two 90° elbows, same plane | 28D |
| Two 90° elbows, perpendicular planes | 54D |
| Gate valve, fully open | 18D |
| Control valve, any position | 50D — avoid if possible |
| Downstream (any fitting) | 5D minimum |
For steady, incompressible flow, mass flow rate is the same at every cross section. Since density ρ is constant, volumetric flow rate Q is also constant:
From this: v₁ = v₂ · (A₂/A₁) = v₂ · (d/D)² = v₂ · β². So pipe velocity is smaller than orifice velocity by a factor of β². For β = 0.5, the pipe velocity is one-quarter of the orifice velocity.
Apply Bernoulli between section 1 (upstream tap) and section 2 (orifice throat), assuming steady, inviscid, incompressible, horizontal flow:
Rearranging:
Substituting v₁ = β²v₂ from the continuity equation:
Solving for the orifice velocity v₂:
Multiply orifice velocity by orifice area to get the theoretical (ideal) volumetric flow rate:
This is what you would measure if the fluid were ideal (inviscid) and the jet filled the bore exactly. In reality, two physical effects cause actual flow to differ.
Effect 1 — Vena Contracta (jet contraction): Due to fluid inertia, streamlines converge beyond the sharp bore edge. The actual minimum jet cross-section (the vena contracta) is smaller than the bore area by the contraction coefficient Cc. For a sharp-edged orifice, Cc ≈ 0.61, meaning the jet uses only 61% of the bore area. This is the primary reason Cd ≈ 0.61.
Effect 2 — Viscous and turbulent losses: Real fluid has viscosity; energy is dissipated in the shear layer and turbulent mixing zone. A velocity coefficient Cv ≈ 0.98 accounts for this loss.
Combined into one coefficient:
Therefore the actual flow rate is:
The challenge: Cd (via the RG equation) depends on Re_D, and Re_D depends on Q (which depends on Cd). This circular dependency is resolved by iteration:
Start with Cd⁰ = 0.61 (or the high-Re asymptote of the RG equation for the given β).
Compute Q = Cd · E · A₂ · √(2ΔP/ρ), then Re_D = 4ṁ / (π · D · μ) = 4ρQ / (π · D · μ).
Substitute Re_D into the RG equation to get Cd_new.
If |Cd_new − Cd_old| < 0.0001, stop. Otherwise set Cd_old = Cd_new and return to step ii. Typically converges in 3–5 iterations.
For gases, density decreases as pressure drops across the orifice. The fluid that exits at P₂ is less dense than at entry P₁, so actual volumetric flow is higher than the incompressible calculation predicts. The isentropic expansion factor Y corrects for this:
When ΔP/P₁ < 2% (a common situation in process plants at operating pressure), Y > 0.997 and the correction is negligible (<0.3%). When ΔP/P₁ > 5%, Y must be applied. For ΔP/P₁ > 25%, the isentropic assumption breaks down and the full energy equation with real gas properties is needed.
Collect from process datasheet and P&ID:
You need density ρ and dynamic viscosity μ at P₁ and T₁ — not at standard conditions. This is the most common source of large errors.
Never use nominal pipe size. Look up actual ID from ASME B36.10 (carbon steel) or B36.19 (stainless) pipe tables for your schedule. Example: 4" pipe, Schedule 40: ID = 102.26 mm. Schedule 80: ID = 97.18 mm. Using the wrong schedule introduces a β error that propagates as a β⁴ error in the flow equation — easily 5–10% flow error.
For existing plant piping, verify with ultrasonic thickness measurement — corrosion, erosion, or scale may have changed the ID significantly from as-built.
Choose a target ΔP at Q_max — this is your DP transmitter upper range value (URV). Common choices: 25 kPa (100 inH₂O), 50 kPa, 125 kPa. Rearrange the orifice equation to solve for d:
d = D · √β where β is found by solving:
Q_max = Cd · E · (π/4 · β²D²) · √(2ΔP_max/ρ)
Since both Cd and E depend on β, iterate: start with β = 0.6, compute d, calculate E and Cd at this β and Re_D, recompute d, repeat until β converges (usually 3 iterations). Confirm β is in range 0.20–0.75. If outside, adjust DP URV or pipe size.
Minimum flow is the worst case for Reynolds number. Calculate:
Re_D = 4 · ρ · Q_min / (π · D · μ)
If Re_D < 10,000, measurement uncertainty increases significantly. ISO 5167 minimum is 5,000. Below this, consider a quadrant-edge orifice, or a different meter type (Coriolis or electromagnetic) if low-flow accuracy is essential.
Map all fittings within 50D upstream and 10D downstream of the proposed orifice location. Look up required straight run from ISO 5167 Table A.1 (or AGA-3 Table 2) for your β and specific fitting combination. If straight run is inadequate, specify a CPA 50E or equivalent flow conditioner. Note on the datasheet: tap orientation (horizontal for liquids, top for gases, condensate-leg for steam), gasket material and thickness (affects effective bore diameter), and whether a senior orifice fitting is required for in-line plate changes.
| Symptom | Likely Cause | Diagnostic / Corrective Action |
|---|---|---|
| Consistently high reading, 5–15% | Worn/rounded upstream bore edge; reversed plate; low-pressure impulse line partially blocked | Pull plate, measure bore, inspect edge under magnifier; verify plate direction arrow; blow down LP impulse line |
| Consistently low reading, 5–15% | Partial bore blockage (debris, scale, wax); condensate in gas LP impulse line; blocked HP tap | Pull and clean plate; drain gas impulse lines; clear HP tap with probe rod or high-pressure N₂ purge |
| Erratic, noisy DP signal | Two-phase flow (flashing or wet gas); pulsation from reciprocating machines; partial tap blockage causing slug flow in impulse line | Check process conditions for flash point; install pulsation dampener or snubber; blow down both impulse lines simultaneously |
| Zero reading at known flow | Manifold in equalise position; both impulse lines blocked or cross-connected; DP transmitter failed low | Check 5-valve manifold configuration; trace both impulse lines physically; verify transmitter with hand pump |
| Negative DP reading | HP and LP connections swapped at transmitter or at taps | Swap HP/LP at transmitter first (safe); if still negative, re-identify which tap is actually upstream on pipe |
| Reading drifts slowly over months | Gradual bore erosion changing d and β; DP transmitter drift; slow impulse line fouling | Annual bore measurement in dirty service; recalibrate transmitter every 2 years; fit isolation valves for blowdown without process shutdown |
| Step change in reading after maintenance | Plate installed backwards; wrong plate (different β) reinstalled; gasket protruding into bore; impulse lines reconnected backwards | Pull plate and verify β marking matches datasheet; check that gasket ID ≥ pipe ID; verify HP/LP connections per P&ID |
Because the Bernoulli equation relates pressure difference to the square of velocity: ΔP = ½ρv². Solving for v gives v ∝ √ΔP, and since Q = Av, we have Q ∝ √ΔP. This is a direct consequence of kinetic energy being proportional to v².
The practical consequence is poor turndown. If maximum flow gives ΔP_max = 100 kPa:
This limits orifice meters to a practical turndown of 3:1 to 5:1, compared to 100:1 for Coriolis meters. For applications requiring wider flow range (e.g., variable-flow processes), consider using two transmitters with different ranges on the same taps, or specifying a senior orifice fitting with interchangeable bore plates.
When fluid approaches a sharp-edged orifice, the streamlines must turn sharply inward to pass through the bore. Due to fluid inertia (momentum), the streamlines cannot make a 90° turn at the bore edge — they overshoot and continue converging beyond the bore face. The jet cross-section continues to decrease for a short distance downstream until reaching a minimum area at the vena contracta ("contracted vein" in Latin). Beyond that point, the jet expands and turbulent mixing with the surrounding slow fluid begins.
The ratio of vena contracta area to bore area is the contraction coefficient Cc. For a sharp-edged orifice, the streamlines' inability to negotiate the abrupt 90° corner results in Cc ≈ 0.611 — a result that can be derived analytically for a 2D slot using complex potential flow theory (the Kirchhoff solution) and is well confirmed experimentally for 3D orifices.
Since Cd = Cc × Cv and Cv ≈ 0.98 (small viscous loss correction), Cd ≈ 0.61 × 0.98 ≈ 0.60. The 0.61 figure is not arbitrary — it is fundamentally the geometric consequence of fluid inertia at a sharp-edged opening.
For a rounded-entry orifice, streamlines follow the curved surface smoothly with no separation, Cc → 1.0, and Cd → 0.98. This is why flow nozzles (rounded entry) have Cd ≈ 0.96–0.99.
The orifice equation shows Q ∝ √(1/ρ). If density changes at the same ΔP, actual flow changes. In automated flow computers (RTUs, DCS), pressure and temperature are read continuously and used to compute flowing density in real time — this is called flow compensation and is mandatory for custody transfer metering.
For manual correction when conditions deviate from design:
Q_actual = Q_indicated × √(ρ_design / ρ_actual)
Examples:
The permanent pressure loss ΔP_perm is the irrecoverable portion of ΔP — the energy permanently converted to heat by turbulent mixing downstream of the vena contracta. It is approximately:
ΔP_perm ≈ ΔP × (1 − β²) / (1 + β²) [sharp-edged orifice, approximate]
For β = 0.5: ΔP_perm ≈ ΔP × 0.60 (60% lost). For β = 0.7: ΔP_perm ≈ ΔP × 0.34 (34% lost). Compare to a venturi tube where permanent loss ≈ 5–10% of ΔP.
Annual pumping cost of permanent pressure loss:
Cost ($/yr) = (ΔP_perm × Q × 8760 × C_elec) / (η_pump × 3,600,000)
ISO 5167-2 is the international standard covering orifice plates for liquids, gases, and steam. It allows corner taps, flange taps, and D&D/2 taps. Uses the RG Cd equation. Published by ISO, adopted in Europe, Asia, and international projects. Applies to all fluids.
AGA Report No. 3 (ANSI/API 14.3) is the North American standard specifically for natural gas custody transfer. It mandates flange taps, uses the RG Cd equation (harmonised with ISO 5167 in 1992), requires gas compressibility calculated per AGA-8, and specifies additional quality requirements for custody transfer including flow computer standards, chart integration, and uncertainty budgets.
For most engineering calculations, the two give essentially identical results (<0.1% difference) when using the same tap configuration and RG equation. Use AGA-3 if: you are metering natural gas for billing/custody transfer in North America, or the contract specifies it. Use ISO 5167 if: you are outside North America, metering liquids or steam, or the contract/local regulations specify it. For process indication (non-custody), either standard is acceptable — just be consistent.
Orifice plates are a poor choice in the following situations:
For a standard orifice plate conforming to ISO 5167, the combined uncertainty is approximately ±0.5% to ±2% of reading. The main contributors are:
For custody transfer applications, an uncertainty budget per ISO 5167 Annex B or AGA-3 is required, documenting each contributor with its Type A or Type B evaluation.