Orifice Flow Calculator

An orifice plate creates a measurable pressure drop that maps to flow:

$Q = C_d\,E\,A_0\sqrt{\frac{2\,\Delta P}{\rho}}, \qquad E = \frac{1}{\sqrt{1-\beta^4}}, \qquad \beta = \frac{d}{D}$

Symbols and SI units:

• $Q$ — volumetric flow, m³/s
• $C_d$ — discharge coefficient, dimensionless; typically $0.60$–$0.62$ for a sharp-edged concentric plate
• $E$ — velocity-of-approach factor, dimensionless
• $A_0 = \pi d^2/4$ — orifice bore area, m²; $d$ bore, $D$ pipe internal diameter, m
• $\Delta P$ — differential pressure across the taps, pascals (Pa)
• $\rho$ — fluid density, kg/m³

For free discharge from a tank under a head $h$ (m), $\Delta P/\rho = gh$ and the equation reduces to $Q = C_d A_0 \sqrt{2gh}$.

The assumption people forget: the flow is treated as incompressible. For gases you must add the expansibility factor $\varepsilon$, and $C_d$ itself depends on Reynolds number, $\beta$ and tap arrangement.

Because $Q \propto \sqrt{\Delta P}$, the relationship is strongly non-linear: halving the flow quarters the differential, which is why an orifice meter has a usable turndown of only about 3:1.

To size a bore, fix a target differential at maximum flow and solve for $A_0$:

$A_0 = \frac{Q}{C_d E \sqrt{2\Delta P/\rho}}, \qquad d = \sqrt{\frac{4A_0}{\pi}}$

Keep $\beta$ within roughly $0.2$–$0.75$; outside that band the coefficients are not well characterised.

The measured $\Delta P$ is not the permanent loss. Some pressure recovers downstream, and the unrecovered loss is roughly $(1-\beta^{1.9})\Delta P$ — around $70\%$ of the differential at $\beta = 0.5$. That loss is a permanent pumping cost.

The rule that governs the real installation: a metering orifice must be designed and installed to ISO 5167 (or an equivalent standard such as AGA 3 for gas), which fixes the plate geometry, tap positions, straight-run lengths and the $C_d$ correlation. The hand calculation here shows the method; it is not a substitute for the standard.

• Dropping the velocity-of-approach factor — at $\beta = 0.5$, $E = 1.033$, and it grows quickly for larger $\beta$.
• Mixing pressure units — $\Delta P$ must be in pascals. $20$ kPa is $20{,}000$ Pa; a bar is $100{,}000$ Pa.
• Assuming flow is proportional to $\Delta P$ — it goes with the square root, so a differential-pressure transmitter output must be square-rooted before it means flow.
• Treating the measured differential as the permanent loss — much of it recovers downstream.
• Using a liquid $C_d$ for compressible gas flow — the expansibility factor is required, and choked flow needs a different treatment entirely.
• Ignoring upstream straight run — an elbow too close to the plate biases the reading regardless of how good the arithmetic is.