Flowmeter

Calculate flow rate and velocity for orifice plate, Venturi tube, and Pitot tube flowmeters

Inputs

d

Orifice diameter d

\Delta P

Pressure difference ΔP

p₁ − p₂ for orifice/Venturi; p₂ − p₁ (stagnation − static) for Pitot

\rho

Fluid density ρ

kg/m³

C

Discharge coefficient C

Typically 0.60–0.65 for standard orifice plates

Formula Interpretation

Orifice Plate Flow

Q = CA\sqrt{\dfrac{2\Delta P}{\rho}} = CA\sqrt{2gH}

Orifice area $A$ computed from orifice diameter $d$ . $C$ accounts for vena contracta and velocity effects. $H$ is the equivalent head of the pressure difference.

H = \dfrac{\Delta P}{\rho g}

$g = 9.81\,\text{m/s}^2$

Key Concepts

Bernoulli Principle

All three meters are based on Bernoulli's equation: a reduction in flow area increases velocity and decreases pressure, enabling flow measurement from the pressure difference.

Discharge Coefficient C

C accounts for real-fluid effects (friction, flow contraction). Orifice plates have lower C (~0.6) due to the abrupt contraction; Venturi tubes approach C ≈ 1 with gradual geometry.

Head Difference H

H = ΔP/(ρg) converts pressure difference to an equivalent liquid column height. With a manometer of liquid density ρ′: H = (ρ′/ρ − 1)H′ for orifice/Venturi; H = (ρ′/ρ)H′ for Pitot.

Worked Example

An orifice plate with orifice diameter $d = 40\,\text{mm}$ is installed in a water pipe. The pressure difference across the plate is $\Delta P = 49100\,\text{Pa}$ . Given discharge coefficient $C = 0.6$ and water density $\rho = 1000\,\text{kg/m}^3$ , find the volumetric flow rate.

d = 40 mm, ΔP = 49 100 Pa, C = 0.6, ρ = 1000 kg/m³

Step 1 — Head difference

H = \dfrac{\Delta P}{\rho g} = \dfrac{49100}{1000\times9.81} = 5.01\,\text{m}

Step 2 — Orifice area

A = \dfrac{\pi}{4}d^2 = \dfrac{\pi}{4}\times(0.04)^2 = 1.257\times10^{-3}\,\text{m}^2

Step 3 — Flow rate

Q = CA\sqrt{2gH} = 0.6\times1.257\times10^{-3}\times\sqrt{2\times9.81\times5.01} \approx 7.47\times10^{-3}\,\text{m}^3/\text{s}

Flow rate Q ≈ 0.00747 m³/s = 7.47 L/s

Extended Knowledge

•Orifice plates are the cheapest option but have the highest permanent pressure loss (~50–70 % of ΔP). Venturi tubes recover most of the pressure (permanent loss < 10 %).
•The square-root relationship between ΔP and Q means that at 50 % of maximum flow, the differential pressure is only 25 % of its maximum value — accuracy is lower at low flow rates.
•Pitot tubes measure only point velocity. An averaging Pitot tube (annubar) samples multiple points across the pipe cross-section to give a better estimate of mean velocity.
•For compressible gas flows, an expansion factor Y is added: Q = CYA√(2ΔP/ρ), where Y < 1 accounts for gas density changes through the meter.