Orifice Flow Meter Theory
What is an Orifice Flow Meter?
An orifice flow meter is a differential pressure (DP) device used to measure volumetric or mass flow rate of fluids (liquids, gases, or steam). It consists of a thin plate with a precisely sized circular hole (bore) installed perpendicular to the flow direction in a pipe.
As fluid flows through the orifice, the restriction causes an increase in velocity and a corresponding decrease in pressure. By measuring the differential pressure (ΔP) across the orifice plate, the flow rate can be calculated using the Bernoulli equation and empirical discharge coefficient correlations.
Operating Principle - Bernoulli Equation
The orifice flow meter operates on the Bernoulli principle, which states that for incompressible, frictionless flow, the sum of pressure energy, kinetic energy, and potential energy remains constant:
P₁/ρ + v₁²/2 + gz₁ = P₂/ρ + v₂²/2 + gz₂
For horizontal flow (z₁ = z₂) with continuity equation (A₁v₁ = A₂v₂), this simplifies to:
ΔP = P₁ - P₂ = (ρ/2)(v₂² - v₁²)
The velocity at the orifice (v₂) is higher than in the pipe (v₁), creating a measurable pressure drop proportional to the square of flow rate.
ISO 5167 Flow Equation
The ISO 5167 standard provides the fundamental equation for orifice flow meters:
Q = Cd × E × A × Y × √(2ΔP/ρ)
where:
- • Q = volumetric flow rate (m³/s)
- • Cd = discharge coefficient (dimensionless)
- • E = velocity of approach factor = 1/√(1-β⁴)
- • A = orifice area = π d²/4 (m²)
- • Y = expansion factor (≈1 for liquids)
- • ΔP = differential pressure (Pa)
- • ρ = fluid density (kg/m³)
Beta Ratio (β)
The beta ratio is the fundamental geometric parameter:
β = d/D
where d = orifice diameter, D = pipe diameter
| Beta Range | Characteristics | Applications |
|---|---|---|
| 0.2 - 0.4 | High ΔP, high accuracy, high permanent loss | Low flow rates, high precision needed |
| 0.4 - 0.6 | Balanced ΔP and permanent loss | Most common industrial applications |
| 0.6 - 0.75 | Low ΔP, lower accuracy, low permanent loss | High flow rates, energy conservation important |
Discharge Coefficient (Cd)
The discharge coefficient accounts for real flow effects including:
- Flow contraction at the vena contracta (minimum flow area)
- Friction losses at the orifice edge
- Velocity profile non-uniformity
- Reynolds number effects
ISO 5167-2 provides the Reader-Harris/Gallagher equation for corner-tapped orifices:
Cd = 0.5961 + 0.0261β² - 0.216β⁸ + 0.000521(10⁶β/ReD)⁰·⁷
+ [0.0188 + 0.0063(19000β/ReD)⁰·⁸] × β⁴/(1-β⁴)
Typical values: Cd = 0.60 - 0.62 for most applications.
Reynolds Number Requirements
The pipe Reynolds number is critical for correlation validity:
ReD = ρvD/μ = 4Q/(πDμ)
Minimum Reynolds Number:
- ReD ≥ 5,000 for β < 0.6 (corner taps)
- ReD ≥ 10,000 for β ≥ 0.6
- For flange taps: ReD ≥ 5,000 minimum
- Below minimum: Discharge coefficient uncertainty increases
Expansion Factor (Y)
The expansion factor corrects for gas/vapor compressibility:
- Liquids: Y ≈ 1.0 (incompressible)
- Gases: Y depends on pressure ratio and isentropic exponent (k)
- Steam: Y calculated from steam properties
For ideal gases (ISO 5167):
Y = 1 - (0.41 + 0.35β⁴)(ΔP/kP₁)
Permanent Pressure Loss
Unlike the recoverable differential pressure, permanent pressure lossrepresents irreversible energy dissipation due to turbulence and friction.
ΔP_perm = K × (ρv²/2)
where K = pressure loss coefficient
Typical permanent loss is 40-90% of differential pressure, depending on β:
- β = 0.5: ~70% of ΔP is permanent loss
- β = 0.7: ~50% of ΔP is permanent loss
- Lower β → Higher permanent loss (more energy wasted)
Pressure Tap Locations
ISO 5167 defines three standard tap configurations:
Corner Taps
Immediately adjacent to orifice plate faces. Most common in Europe. Compact, less sensitive to pipe diameter.
Flange Taps
1 inch upstream and downstream of plate faces. Common in North America. Easy installation.
D and D/2 Taps
1D upstream, 0.5D downstream. Used for research and special applications.
Installation Requirements
Proper installation is critical for accurate measurement:
Straight Pipe Requirements:
- Upstream: Minimum 10-44D depending on fitting type and β
- Downstream: Minimum 4-8D
- Flow conditioners may reduce straight pipe requirements
- Two 90° elbows in same plane: Up to 44D upstream
- Two 90° elbows in perpendicular planes: Up to 34D upstream
- Concentricity: Orifice bore must be concentric with pipe (within 0.005D)
- Edge sharpness: Upstream edge must be sharp (critical for Cd)
- Plate thickness: Typically E = 0.005D to 0.02D
- Surface roughness: Ra < 0.0001D on upstream face
Advantages and Limitations
Advantages
- ✓ Simple, reliable, no moving parts
- ✓ Low initial cost
- ✓ Well-established ISO 5167 standard
- ✓ Wide range of applications
- ✓ High temperature and pressure capability
- ✓ Easy to inspect and replace
- ✓ Suitable for large pipe sizes
Limitations
- ✗ Significant permanent pressure loss
- ✗ Non-linear relationship (√ΔP)
- ✗ Limited rangeability (3:1 typical)
- ✗ Requires straight pipe runs
- ✗ Sensitive to erosion and fouling
- ✗ Not suitable for slurries or dirty fluids
- ✗ Accuracy decreases at low Reynolds numbers
Design Example
Problem: Size an orifice meter for water
• Pipe diameter: D = 100 mm
• Flow rate: Q = 50 m³/h = 0.0139 m³/s
• Water density: ρ = 1000 kg/m³
• Viscosity: μ = 0.001 Pa·s
• Beta ratio: β = 0.5
Solution:
1. Orifice diameter: d = β × D = 0.5 × 100 = 50 mm
2. Pipe velocity: v = Q/A = 0.0139/(π×0.05²) = 1.77 m/s
3. Reynolds number: Re = ρvD/μ = 1000×1.77×0.1/0.001 = 177,000 (✓ turbulent)
4. Discharge coefficient: Cd ≈ 0.608 (from ISO 5167 correlation)
5. Differential pressure: ΔP ≈ 5.8 kPa
6. Permanent loss: ΔP_perm ≈ 3.5 kPa (60% of ΔP)
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"
- Reader-Harris, M.J., "Orifice Plates and Venturi Tubes", Springer (2015)
- ASME MFC-3M-2004, "Measurement of Fluid Flow in Pipes Using Orifice, Nozzle, and Venturi"
- Miller, R.W., "Flow Measurement Engineering Handbook", 3rd Edition, McGraw-Hill (1996)
- AGA Report No. 3, "Orifice Metering of Natural Gas and Other Related Hydrocarbon Fluids", American Gas Association (2013)