Introduction
A restriction orifice (RO) — sometimes called a flow restrictor, fixed orifice, or orifice plate — is a plate with a precisely drilled hole installed in a piping flange pair. Unlike a control valve or pressure regulator, it has no moving parts. It presents a fixed restriction to flow, and the process conditions determine the resulting pressure drop and flow rate at any given moment.
Restriction orifices are among the most common items in an upstream process system. They appear on:
- Compressor recycle lines — ensuring a minimum flow through the compressor at all times, protecting against surge even if the recycle control valve fails closed.
- Bypass and equalisation lines — limiting the flow that can pass through a manual bypass, preventing an operator from inadvertently overloading downstream equipment.
- Chemical and inhibitor injection — balancing flow between parallel injection points or capping the maximum injection rate.
- Pressure letdown — providing a fixed, passive pressure reduction on low-flow lines where a control valve would be disproportionate.
- Flare and blowdown systems — controlling peak flare loads during depressurisation.
Despite their simplicity, restriction orifices are frequently sized incorrectly. An undersized RO fails to achieve its design intent. An oversized one creates excessive noise, vibration, and — in liquid service — cavitation erosion that can destroy the orifice and the surrounding pipework within months. The sizing calculation is not complicated, but it requires care and an understanding of the fluid mechanics involved.
Liquid Service — The Basic Equation
For single-phase incompressible (liquid) flow through a sharp-edged orifice, the governing equation is derived from Bernoulli's theorem with an empirical discharge coefficient to account for real-fluid effects at the orifice throat:
Q = Cd × A₀ × √(2 × ΔP / ρ)
Where:
- Q — volumetric flow rate (m³/s)
- Cd — discharge coefficient (dimensionless)
- A₀ — orifice area = π/4 × d² (m²)
- ΔP — differential pressure across the orifice (Pa)
- ρ — liquid density at process conditions (kg/m³)
Rearranging for the orifice diameter d given a required flow and available pressure drop:
d = √[ 4Q / (π × Cd × √(2 × ΔP / ρ)) ]
Discharge coefficient: For a sharp-edged orifice with a beta ratio (β = d/D, where D is the pipe bore) in the range 0.2 to 0.75, the discharge coefficient Cd is approximately 0.61 per ISO 5167. This value is well-established and applies across a wide range of Reynolds numbers in turbulent flow. For very small orifices or low Reynolds number conditions, Cd can deviate — consult ISO 5167 Part 2 for the full correlation.
Cavitation — The Check You Cannot Skip
For any liquid-service restriction orifice, the cavitation check is mandatory. This is the step most frequently omitted in practice, and its omission is the primary cause of premature orifice failure.
As liquid accelerates through the orifice throat, its static pressure drops sharply. At the vena contracta — the point of minimum flow area just downstream of the orifice plate — the local pressure reaches its minimum value. If this minimum pressure falls below the vapour pressure of the liquid at process temperature, the liquid partially vapourises. When this vapour then collapses as the flow decelerates and pressure recovers downstream, it does so violently — the collapse of vapour bubbles produces shock waves that pit and erode the orifice bore and the adjacent pipework. This is cavitation.
The cavitation index σ provides a simple screening check:
σ = (P₁ - Pv) / ΔP
Where:
- P₁ — upstream absolute pressure (Pa)
- Pv — vapour pressure of the liquid at process temperature (Pa)
- ΔP — differential pressure across the orifice (Pa)
If σ < 1.7 (a conservative threshold for standard sharp-edged orifices), cavitation should be anticipated and the design modified.
Remedies for cavitation risk:
Staged orifices in series — split the total pressure drop across two or more orifice plates in series, with sufficient pipe length between them to allow pressure recovery above vapour pressure before the next stage.
Increase downstream back-pressure — raising the system back-pressure increases P₁ relative to Pv for the same ΔP, improving the cavitation index. This is sometimes achievable by relocating the orifice to a different position in the system.
Anti-cavitation trim — proprietary multi-stage orifice assemblies (similar in principle to anti-cavitation control valve trim) that dissipate pressure energy in a series of small steps rather than one large drop.
Gas Service — Compressible Flow
For gas or vapour service, the fluid density changes as it accelerates through the orifice and the Bernoulli approach must be modified. The first and most important determination is whether the flow is subcritical (subsonic) or critical (choked at the orifice throat).
Critical Flow Condition
Flow through an orifice reaches the critical (choked) condition when the downstream pressure P₂ falls below the critical pressure ratio:
P₂ / P₁ ≤ [2 / (k + 1)]^[k / (k - 1)]
Where k is the isentropic exponent (specific heat ratio Cp/Cv) of the gas. For typical natural gas (predominantly methane), k ≈ 1.31. For heavier hydrocarbon gas mixtures, k is lower — typically 1.10 to 1.25. For k = 1.25:
Critical pressure ratio = [2 / 2.25]^(1.25/0.25) = 0.889^5 ≈ 0.556
In most pressure letdown, blowdown, and flare bypass applications, the pressure ratio across the orifice is large enough that critical flow prevails. Once the flow is choked, further reduction in downstream pressure does not increase the mass flow rate — only upstream conditions determine throughput.
Mass Flow at Critical Conditions
ṁ = Cd × A₀ × P₁ × √[ k × M / (R × T₁) × (2 / (k + 1))^((k + 1) / (k - 1)) ]
Where:
- ṁ — mass flow rate (kg/s)
- M — molar mass of the gas (kg/kmol)
- R — universal gas constant (8314 J/kmol·K)
- T₁ — upstream temperature (K)
This equation assumes isentropic flow and an ideal gas. For real gases at high pressure, a compressibility factor Z should be incorporated into the upstream density calculation.
Subcritical Gas Flow
When the pressure ratio is above the critical value (i.e. the downstream pressure is relatively high), the ISO 5167 expansion factor Y is applied:
ṁ = Cd × Y × A₀ × √(2 × ρ₁ × ΔP)
Where Y is calculated from the pressure ratio τ = P₂/P₁ and the isentropic exponent k per ISO 5167. For preliminary calculations, Y ≈ 1 − (0.41 + 0.35 × β⁴) × ΔP / (k × P₁) provides a reasonable approximation.
Beta Ratio — Practical Constraints
The beta ratio β = d/D should be maintained within 0.2 ≤ β ≤ 0.75 for the ISO 5167 discharge coefficient correlation to apply with confidence.
β < 0.2: The orifice is very small relative to the pipe. This is geometrically acceptable — the pipe bore is not a constraint in the way it is for a measurement orifice — but it produces very high velocities through the orifice throat. In gas service this can cause acoustic noise and fatigue. In liquid service with suspended solids or any two-phase flow, it accelerates erosion.
β > 0.75: The restriction effect is minimal and the differential pressure generated is small. The orifice is also more difficult to dimension and install with the precision needed to achieve a predictable Cd.
Where the required duty produces a β below 0.2, staged orifices in series should be considered as the preferred solution.
Multiple Orifices in Series
Staging is the standard approach when a single-stage design would result in cavitation (liquid service) or unacceptable noise levels (gas service). The total pressure drop is distributed across two or more stages, with the pipe length between stages sufficient to allow pressure recovery and phase stabilisation.
The sizing procedure for staged orifices is iterative:
- Assume an intermediate pressure between stages
- Size each stage independently using the single-orifice equations
- Verify that the intermediate pressure prevents cavitation (liquid) or keeps the upstream stage subcritical if that is the design intent (gas)
- Adjust the assumed intermediate pressure and repeat until consistent
For two-stage liquid ROs, a practical starting assumption is to split the total pressure drop equally between stages, then adjust to achieve σ > 2.0 at each stage.
A Worked Example — Liquid Service
Given:
- Fluid: produced water at 60°C
- Density ρ: 995 kg/m³
- Vapour pressure Pv: 0.20 bar a (20 000 Pa)
- Required flow Q: 5.0 m³/h = 0.00139 m³/s
- Upstream pressure P₁: 12.0 bar a (1 200 000 Pa)
- Downstream pressure P₂: 4.0 bar a (400 000 Pa)
- ΔP: 8.0 bar = 800 000 Pa
- Pipe bore D: 50 mm = 0.050 m
- Cd: 0.61
Step 1 — Calculate orifice area and diameter:
A₀ = Q / (Cd × √(2 × ΔP / ρ))
= 0.00139 / (0.61 × √(2 × 800 000 / 995))
= 0.00139 / (0.61 × √1608)
= 0.00139 / (0.61 × 40.1)
= 0.00139 / 24.46
= 5.68 × 10⁻⁵ m²
d = √(4 × A₀ / π) = √(4 × 5.68 × 10⁻⁵ / π) = √(7.24 × 10⁻⁵) = 8.5 mm
Beta ratio check: β = 8.5 / 50 = 0.17 — below the 0.20 lower bound. Consider a staged approach or accept a slightly higher Cd uncertainty for a restriction (non-measurement) application.
Step 2 — Cavitation check:
σ = (P₁ - Pv) / ΔP = (1 200 000 - 20 000) / 800 000 = 1.475
σ = 1.47 < 1.7 — cavitation is a concern. Consider two stages with ΔP split 400 000 Pa each, intermediate pressure ≈ 8.0 bar a.
σ_stage1 = (1 200 000 - 20 000) / 400 000 = 2.95 ✓
σ_stage2 = (800 000 - 20 000) / 400 000 = 1.95 ✓ (marginal — check detailed)
Two stages resolve the cavitation concern. Each stage is sized independently using the single-orifice formula with its own ΔP.
Key Standards
- ISO 5167 — the primary reference for orifice plate geometry, installation requirements, and discharge coefficient correlations. Written for measurement orifices, but the hydraulic relationships and Cd values apply directly to sharp-edged restriction orifices of the same geometry.
- API 520 / API 521 — applicable when the RO forms part of a relief or depressurisation system.
- ASME B16.36 — orifice flanges and associated fittings.
Summary
Restriction orifices are simple in concept and frequently treated as minor items. They are not minor in consequence. An incorrectly sized RO on a compressor recycle can drive the machine into surge. One on a chemical injection header can cause maldistribution and under-dosing across an entire injection system. One in liquid service without a cavitation check can require pipework replacement within a year of start-up.
Size them rigorously, check for cavitation in liquid service, check for critical flow in gas service, and verify the beta ratio. The calculation takes less time than the consequences of getting it wrong.
