Orifice Flow Formula — ISO 5167 & AGA-3 Discharge Coefficient

An orifice plate is a thin plate with a precisely machined concentric bore inserted between two pipe flanges. Flow through the orifice creates a measurable differential pressure that scales as the square of the volumetric flow rate. ISO 5167 (international) and AGA Report No. 3 (U.S. custody-transfer standard) define the discharge coefficient, geometry tolerances, and installation requirements that make the measurement reproducible to within 0.5% for sized installations.

Overview

An orifice differential-pressure (DP) meter is a head-type flow meter. The plate creates a localized contraction (vena contracta) where velocity is highest and pressure is lowest. By tapping pressure upstream and downstream, the meter measures the head differential, which Bernoulli's equation relates to velocity, and through pipe area, to volumetric flow. The discharge coefficient C_d corrects the ideal-fluid Bernoulli equation for real-fluid effects (boundary layer, vena contracta location, jet contraction).

Theory

Bernoulli's incompressible-flow energy balance across an orifice gives an ideal velocity of sqrt(2*ΔP/ρ). The discharge coefficient C_d empirically corrects for actual streamline contraction and friction loss, and the expansibility factor Y corrects for gas density changes across the orifice. The Reader-Harris/Gallagher (RG) equation is the canonical Cd model used in both ISO 5167-2 and AGA Report No. 3.

Formulas

General Orifice Flow Equation

Q = Cd * Y * A_orifice * sqrt(2 * deltaP / rho) / sqrt(1 - beta^4)

A_orifice = (pi/4) * d^2
beta      = d / D

d = orifice bore diameter, D = pipe inside diameter, beta = beta ratio (typical 0.20–0.75), ρ = upstream fluid density at flowing conditions, ΔP = measured differential pressure, Y = expansibility factor (Y = 1 for incompressible liquids).

Reader-Harris / Gallagher Cd (ISO 5167-2, AGA-3)

Cd = 0.5961 + 0.0261*beta^2 - 0.216*beta^8
   + 0.000521 * (1e6 * beta / Re_D)^0.7
   + (0.0188 + 0.0063*A) * beta^3.5 * (1e6 / Re_D)^0.3
   + (0.043 + 0.080*exp(-10*L1) - 0.123*exp(-7*L1)) * (1 - 0.11*A) * beta^4 / (1 - beta^4)
   - 0.031 * (M2_prime - 0.8*M2_prime^1.1) * beta^1.3

A         = (19000 * beta / Re_D)^0.8
M2_prime  = 2 * L2_prime / (1 - beta)
L1, L2'   = tap-spacing parameters (flange / corner / D-D/2)

Expansibility (Gas Expansion Factor)

Y = 1 - (0.351 + 0.256*beta^4 + 0.93*beta^8) * (1 - (P2/P1)^(1/kappa))

P₁, P₂ = upstream and downstream tap pressures, κ = isentropic exponent. For liquids, Y = 1. For natural gas, Y typically 0.97–1.00.

Reynolds Number Check

Re_D = (rho * V * D) / mu

V = 4 * Q / (pi * D^2)

Inverting for Differential Pressure or Bore Diameter

deltaP = (Q / (Cd * Y * A_orifice))^2 * rho / 2 * (1 - beta^4)

d = solve via iteration (Cd depends on beta which depends on d)

Key Symbols

Symbol	Description	Units
Q	Volumetric flow rate	m³/s, ft³/s
Cd	Discharge coefficient	dimensionless
Y	Expansibility (gas)	dimensionless
beta	d / D	dimensionless
d	Orifice bore diameter	in, mm
D	Pipe inside diameter	in, mm
deltaP	Differential pressure	inH2O, kPa
rho	Upstream density (flowing)	lb/ft³, kg/m³

Worked Example

Given: Water at 60 °F, D = 4 in pipe, d = 2 in orifice (beta = 0.5), ΔP = 50 inH2O, Cd = 0.605 (Re_D ~ 5e5 from iteration).

Step 1 — Areas and beta:

A_orifice = (pi/4) * (2/12)^2 = 0.02182 ft^2
beta = 2 / 4 = 0.5
1 - beta^4 = 1 - 0.0625 = 0.9375

Step 2 — Convert deltaP to lbf/ft^2:

deltaP = 50 inH2O * 5.2023 lbf/ft^2 / inH2O = 260.1 lbf/ft^2
rho_water = 62.43 lb/ft^3

Step 3 — Volumetric flow rate:

Q = 0.605 * 1.0 * 0.02182 * sqrt(2 * 260.1 * 32.174 / 62.43) / sqrt(0.9375)
  = 0.0132 * sqrt(268.0) / 0.968
  = 0.0132 * 16.37 / 0.968
  = 0.223 ft^3/s
  = 100.0 GPM

Valid Ranges (ISO 5167 / AGA-3)

Parameter	Range
Pipe ID (D)	50 mm to 1000 mm (2 to 40 in)
Beta ratio	0.10 ≤ beta ≤ 0.75
Re_D (pipe Reynolds)	≥ 5000 (for beta < 0.45); ≥ 170 * beta^2 * D / 25.4 (general)
Upstream straight run	10D minimum, 20D typical
Downstream straight run	5D minimum
Plate thickness	0.005D to 0.02D

When the formula does not apply

Multi-phase flow (gas + liquid) — use a wet-gas meter or separate before measuring.
Inadequate upstream straight run (elbows, valves, tees within 10D upstream) — use a flow conditioner or specify a non-orifice meter.
High beta (> 0.75) — accuracy drops, plate damage risk rises.
Very low Reynolds (< 5000) — transitional flow regime; use a venturi or coriolis meter.
High temperatures or corrosive service — mechanical changes to the plate void the Cd correlation; use an integral plate or wedge meter.

References

ISO 5167-2:2003 — Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full — Part 2: Orifice plates.
AGA Report No. 3 (API MPMS 14.3) — Orifice Metering of Natural Gas and Other Related Hydrocarbon Fluids.
Reader-Harris, M.J. (2015). Orifice Plates and Venturi Tubes. Springer Experimental Fluid Mechanics.
Miller, R.W. (1996). Flow Measurement Engineering Handbook, 3rd ed. McGraw-Hill.
Crane Technical Paper 410 — Flow of Fluids Through Valves, Fittings, and Pipe.

Orifice Flow Formula — ISO 5167 & AGA Report No. 3