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Hagen-Poiseuille Flow Rate Calculator

Calculate the volumetric flow rate of a laminar, viscous fluid flow through a pipe via the Hagen-Poiseuille equation, Q = π·ΔP·r⁴ ÷ (8·μ·L), from pipe diameter, length, pressure drop, and fluid viscosity.

Flow rate (L/min)
2.3562 L/min
Flow rate (US gpm)
0.62244 gpm
Flow rate (m³/s)
0.00003927 m³/s
Average velocity
12.5 m/s
Centerline (max) velocity
25 m/s

How it works

The Hagen-Poiseuille equation gives the flow rate of a steady, laminar flow of a Newtonian fluid through a long, straight pipe: Q = π·ΔP·r⁴ ÷ (8·μ·L), where ΔP is the pressure drop along the pipe, r is the pipe's internal radius, μ is the fluid's dynamic viscosity, and L is the pipe length. It's the viscous counterpart to this site's Venturi Meter calculator, which instead uses Bernoulli's equation for a fast, low-viscosity flow through a sudden constriction.

The radius appears to the 4th power, which makes flow extremely sensitive to pipe size: halving the radius at the same pressure drop cuts the flow rate to 1/16th, not 1/2. Because the flow's velocity profile across the pipe is parabolic (fastest at the center, zero at the wall), the centerline velocity works out to exactly twice the average velocity — reported alongside the flow rate here.

This formula only holds for laminar flow — smooth, layered motion without turbulent mixing. It's the standard model for slow flows of thick fluids (oil, syrup, blood in small vessels) or flow through very narrow tubes (microfluidics), not for fast flow of a thin fluid like water through an ordinary pipe, which is typically turbulent instead.

FAQ

Why does halving the pipe diameter cut the flow so much more than halving the pressure drop does?

Because radius enters the formula to the 4th power while pressure drop enters only linearly. Doubling the pressure drop simply doubles the flow, but doubling the radius increases flow 16-fold (2⁴) — which is exactly why even a little bit of narrowing (from, say, deposits inside a pipe or a vessel) chokes flow far more than you'd intuitively expect.

What happens if I plug in water flowing through a normal-sized pipe at a realistic speed?

You'll get a number out, but it won't be physically accurate — this formula strictly assumes laminar flow, and water through anything but a very narrow tube at a very low speed is almost always turbulent in practice. It's most accurate for thick, slow-moving, or very-small-scale flows; treat results for fast flow of a thin fluid as a rough laminar-flow approximation only.

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