Nozzle Flow Rate Calculator

When a liquid discharges from a nozzle or orifice under pressure into a lower-pressure region (often atmosphere), its exit velocity and flow rate can be calculated from the pressure difference driving the flow, using a form of Bernoulli's equation known as Torricelli's principle:

vᵢᵈₑₐₗ = √(2ΔP/ρ)

This gives the theoretical (ideal, frictionless) exit velocity from the pressure difference ΔP and fluid density ρ. Multiplying that velocity by the nozzle's cross-sectional area gives the theoretical flow rate. Real nozzles, however, don't achieve this ideal velocity exactly — friction, the contraction of the flow stream just past a sharp-edged opening (called the vena contracta), and other real-world effects reduce actual flow below the theoretical figure. This is captured by the discharge coefficient, Cd:

Q = Cd × A × √(2ΔP/ρ)

The discharge coefficient depends heavily on the physical shape of the nozzle or orifice. A simple sharp-edged orifice (a thin plate with a hole) has a relatively low Cd, typically around 0.60–0.65, because the flow separates sharply at the edge and contracts significantly just downstream, reducing the effective flow area well below the geometric hole area. A well-designed, smoothly rounded or converging nozzle profile, by contrast, guides the flow more gently and achieves a Cd much closer to 1.0, typically 0.95–0.99, because the flow stays attached to the nozzle wall all the way to the exit with minimal contraction.

This relationship also shows why flow rate does not scale linearly with pressure — because velocity depends on the square root of ΔP, doubling the upstream pressure increases flow rate by only about 41% (√2), not 100%. This nonlinear, diminishing-returns relationship is an important practical consideration when trying to increase flow through a fixed-size nozzle by increasing pump pressure.

1. Assuming Cd = 1.0 (the ideal, frictionless case). Every real nozzle or orifice has some discharge coefficient below 1.0 — using the ideal velocity directly as if it were the actual flow rate significantly overstates real discharge, especially for a sharp-edged orifice where Cd can be as low as 0.6.

2. Assuming flow rate scales linearly with pressure. Because velocity depends on the square root of ΔP, flow rate increases with the square root of pressure, not proportionally — doubling pressure gives roughly 41% more flow, not 100% more, which trips up quick mental estimates.

3. Using the geometric hole diameter for a sharp-edged orifice's effective flow area. For a sharp-edged orifice, the discharge coefficient already accounts for the flow stream contracting just past the opening (the vena contracta) — don't separately try to estimate a "contracted area" on top of the standard Cd, as that double-counts the effect.

4. Applying this incompressible formula to high-pressure gas or steam nozzles. Gas and steam at significant pressure ratios are compressible and can reach choked (sonic) flow conditions where flow rate stops increasing with further downstream pressure reduction — a completely different set of equations is needed for that regime, not this liquid orifice formula.

5. Using a generic Cd instead of the specific nozzle manufacturer's tested value. Discharge coefficient varies meaningfully with the exact geometry of a nozzle, especially for engineered spray or metering nozzles — use the manufacturer's actual rated Cd or flow-rate-versus-pressure curve for precise applications, treating the generic reference values here as estimates only.

6. Ignoring downstream backpressure. ΔP in this formula is the difference between upstream and downstream pressure, not the upstream (gauge) pressure alone — if the nozzle discharges into a pressurized space rather than atmosphere, only the actual pressure difference drives the flow.