How do I calculate flow through an orifice from the pressure drop?

Use Q = Cd × Ao × √(2ΔP / (ρ(1 − β⁴))), where Ao is the orifice area and β is the orifice-to-pipe diameter ratio. For a 25 mm orifice in a 50 mm line with water and a 50 kPa drop, Cd = 0.61 gives 3.096 L/s.

What discharge coefficient should I use for an orifice plate?

0.61 is a common value for a sharp-edged plate at high Reynolds number. Rounded nozzles and venturis run near 0.95 to 0.99. The true value depends on the Reynolds number, the beta ratio, and the pressure tap positions, so use a calibrated or standard value when accuracy matters.

What is the beta ratio and why does it matter?

Beta is the orifice diameter divided by the pipe diameter, d/D. It sets how much the approaching flow is already moving, through the 1 − β⁴ term. The usual orifice correlations apply for β between about 0.2 and 0.75, and the calculator warns outside that range.

Why does a fourfold pressure drop only double the flow?

Flow is proportional to the square root of the pressure drop. Quadrupling ΔP multiplies Q by √4 = 2. Going the other way, doubling the flow needs four times the pressure drop.

Can I use this for gas flow?

Only when the pressure drop is a small fraction of the line pressure. For larger ratios the gas expands as it accelerates and the incompressible equation overpredicts the flow, so an expansibility factor is needed. ISO 5167 gives the full method.

Orifice Flow Calculator

Flow through an orifice plate or nozzle from differential pressure, orifice and pipe diameter, and discharge coefficient, or the pressure drop for a known flow.

Scratchpad (not saved)

Save to create a persistent, shareable workspace.

Solve For

Orifice Diameter d

Bore of the orifice plate or nozzle throat.

Pipe Inside Diameter D

Inside diameter of the pipe; must be larger than the orifice.

Diameter Unit

Unit for both diameters.

Pressure Drop ΔP

Differential pressure across the orifice. Used when solving for flow.

Pressure Unit

Unit for the value entered above.

Flow Rate Q

Used when solving for pressure drop.

Flow Unit

Unit for the value entered above.

Fluid (Density ρ)

Custom Density ρ

Used only when the fluid is set to Custom.

Custom Density Unit

Unit for the custom density.

Discharge Coefficient Cd

0.61 for a sharp-edged plate. See the table for nozzles and venturis.

What This Calculator Does

A restriction in a pipe converts pressure into velocity, and the pressure difference across it measures the flow. This calculator applies the standard incompressible orifice equation to give the volumetric flow for a known pressure drop, or the pressure drop for a known flow. It uses the orifice diameter $d$, the pipe diameter $D$ through the beta ratio $\beta = d/D$, the fluid density, and a discharge coefficient $C_d$ that defaults to 0.61 for a sharp-edged plate. It warns when $\beta$ falls outside 0.2 to 0.75, where the equation and typical coefficients become unreliable.

It combines Solve For, Orifice Diameter d, Pipe Inside Diameter D, Diameter Unit to estimate Flow rate Q, Pressure drop ΔP, Beta ratio β = d/D.

Formula & Method

Volumetric flow through the orifice:

Q = C_d\,A_o\,\sqrt{\frac{2\,\Delta P}{\rho\left(1 - \beta^4\right)}}, \qquad \beta = \frac{d}{D}, \qquad A_o = \frac{\pi d^2}{4}

Solved for pressure drop:

\Delta P = \frac{\rho\left(1 - \beta^4\right)}{2}\left(\frac{Q}{C_d A_o}\right)^2

where

Q

is volumetric flow,

\Delta P

is the differential pressure across the plate,

\rho

is fluid density,

d

is the orifice bore, and

D

is the pipe inside diameter. The factor

1/\sqrt{1-\beta^4}

accounts for the velocity already present in the approaching pipe flow. It comes from Bernoulli's equation combined with continuity.

Notation used in the formulas: $R$ = Flow rate Q; $x_{1}$ = Solve For; $x_{2}$ = Orifice Diameter d; $x_{3}$ = Pipe Inside Diameter D; $x_{4}$ = Diameter Unit; $x_{5}$ = Pressure Drop ΔP; $x_{6}$ = Pressure Unit.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it to size an orifice for a target flow, estimate flow from a measured pressure drop across a plate or nozzle, or estimate the restriction a flow-limiting orifice puts in a line. Flow scales with the square root of $\Delta P$, so a fourfold increase in pressure drop only doubles the flow.

Worked Examples

Sharp-edged plate in a 50 mm line, water, 50 kPa (metric)

Given: d = 25 mm, D = 50 mm, ΔP = 50 kPa, water ρ = 998 kg/m³, Cd = 0.61 Method: compute β and the orifice area, then apply Q = Cd·Ao·√(2ΔP / (ρ(1 − β⁴))). β = 25/50 = 0.5, so 1 − β⁴ = 0.9375 Ao = π(0.025)² / 4 = 4.909×10⁻⁴ m² √(2 × 50,000 / (998 × 0.9375)) = √106.9 = 10.34 m/s Q = 0.61 × 4.909×10⁻⁴ × 10.34 = 3.096×10⁻³ m³/s = 3.096 L/s Reverse check: with Q = 2 L/s the pressure drop falls to 20.87 kPa, because ΔP scales with Q².

1 in orifice in a 2 in line at 10 psi (imperial)

Given: d = 1 in, D = 2 in, ΔP = 10 psi (68.95 kPa), water, Cd = 0.61 Method: same equation, with diameters converted to metres (0.0254 m and 0.0508 m). β = 0.5, Ao = 5.067×10⁻⁴ m² Q = 0.61 × 5.067×10⁻⁴ × √(2 × 68,948 / (998 × 0.9375)) = 3.752×10⁻³ m³/s Q = 3.75 L/s = 59.5 US gal/min The mean velocity through the orifice bore is 7.41 m/s (24.3 ft/s).

Typical Discharge Coefficients

Approximate ranges from textbook tables for turbulent flow. The coefficient depends on the Reynolds number, $\beta$, tap locations, and edge condition, so a calibrated or standard-specified value is better when accuracy matters.

Restriction	Typical C_d
Sharp-edged orifice plate	0.60 to 0.62
Short tube or square-edged pipe entrance	0.80 to 0.82
Re-entrant (Borda) tube	about 0.50
Well-rounded flow nozzle	0.95 to 0.99
Venturi meter	0.95 to 0.99

Reference

The orifice equation follows from Bernoulli's equation and continuity as presented in Çengel and Cimbala, Fluid Mechanics, and White, Fluid Mechanics. For custody-transfer or regulatory metering, ISO 5167 specifies the geometry, tap locations, and discharge coefficient correlations required.

Common Mistakes

Using the orifice diameter in the beta ratio the wrong way round. $\beta = d/D$, orifice over pipe, so it is always below 1.
Using total pressure instead of the differential pressure across the plate. $\Delta P$ is the difference between the upstream and downstream taps.
Assuming $C_d = 0.61$ for every restriction. A rounded nozzle or venturi has a coefficient near 0.97.
Reading the permanent pressure loss as the measured $\Delta P$. The pressure that is not recovered downstream is a fraction of the measured drop.
Mixing units for pressure and density. A drop in psi with a density in kg/m³ gives nonsense unless converted. Set each unit selector to match its value.

When Not To Use This Calculator

Compressible gas flow with a large pressure ratio. It needs an expansibility factor, and the incompressible equation overpredicts flow.
Custody transfer or regulatory metering. Use the full ISO 5167 procedure, which this calculator does not implement.
Very low Reynolds number or viscous flow, where the discharge coefficient varies strongly.
Partly full pipes, cavitating flow, or two-phase flow.
Beta ratios below 0.2 or above 0.75, where the standard correlations stop applying.

Frequently Asked Questions

How do I calculate flow through an orifice from the pressure drop?: Use Q = Cd × Ao × √(2ΔP / (ρ(1 − β⁴))), where Ao is the orifice area and β is the orifice-to-pipe diameter ratio. For a 25 mm orifice in a 50 mm line with water and a 50 kPa drop, Cd = 0.61 gives 3.096 L/s.
What discharge coefficient should I use for an orifice plate?: 0.61 is a common value for a sharp-edged plate at high Reynolds number. Rounded nozzles and venturis run near 0.95 to 0.99. The true value depends on the Reynolds number, the beta ratio, and the pressure tap positions, so use a calibrated or standard value when accuracy matters.
What is the beta ratio and why does it matter?: Beta is the orifice diameter divided by the pipe diameter, d/D. It sets how much the approaching flow is already moving, through the 1 − β⁴ term. The usual orifice correlations apply for β between about 0.2 and 0.75, and the calculator warns outside that range.
Why does a fourfold pressure drop only double the flow?: Flow is proportional to the square root of the pressure drop. Quadrupling ΔP multiplies Q by √4 = 2. Going the other way, doubling the flow needs four times the pressure drop.
Can I use this for gas flow?: Only when the pressure drop is a small fraction of the line pressure. For larger ratios the gas expands as it accelerates and the incompressible equation overpredicts the flow, so an expansibility factor is needed. ISO 5167 gives the full method.

Inputs Used

Solve For: Used directly in the calculation.
Orifice Diameter d: Bore of the orifice plate or nozzle throat.
Pipe Inside Diameter D: Inside diameter of the pipe; must be larger than the orifice.
Diameter Unit: Unit for both diameters.
Pressure Drop ΔP: Differential pressure across the orifice. Used when solving for flow.
Pressure Unit: Unit for the value entered above.
Flow Rate Q: Used when solving for pressure drop.
Flow Unit: Unit for the value entered above.
Fluid (Density ρ): Used directly in the calculation.
Custom Density ρ: Used only when the fluid is set to Custom.
Custom Density Unit: Unit for the custom density.
Discharge Coefficient Cd: 0.61 for a sharp-edged plate. See the table for nozzles and venturis.

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