How this instrument works
Daniel Bernoulli published the principle behind this calculator in his 1738 treatise Hydrodynamica, describing how a moving fluid's pressure trades off against its speed along a streamline. Applied to a tank draining through a small hole, that trade-off reduces to a clean statement: the pressure pushing the fluid out converts almost entirely into motion at the opening, so a faster exit jet comes directly at the cost of the driving pressure being spent.
PSI and GPM measure two different kinds of quantity — one is force per area, the other is volume per time — so no fixed multiplier links them the way 2.54 links centimetres to inches. Flow through an opening also depends on how large that opening is and how dense the fluid is: double the orifice diameter and its cross-sectional area quadruples, while a denser fluid resists acceleration and exits slower for the same pressure push.
The formula assumes a large reservoir draining through a small orifice or short pipe, so the fluid surface inside the tank is treated as essentially still next to the jet leaving the hole, and the fluid itself is treated as incompressible and frictionless. Real plumbing adds viscosity, turbulence, and pipe-wall roughness working against the flow, so a measured rate typically lands somewhat below this ideal prediction — engineers close that gap with an empirical discharge coefficient, often 0.6 for a sharp-edged orifice up near 0.98 for a smoothly rounded nozzle.
- Enter the driving pressure into the Pressure differential (psi) field — the gap between the pressurized side and the discharge side, defaulted here to 40 psi.
- Set the Pipe/orifice diameter (inches) field to the bore the fluid exits through; area grows with the square of this number, so small changes shift the flow rate a lot.
- Adjust the Fluid density (kg/m^3) field away from the default of 1000 (water) if you're modeling another fluid — lighter fluids exit faster, heavier ones exit slower.
- Exit velocity (m/s) updates first, showing the jet speed straight from Bernoulli's equation before it gets converted into a volume rate.
- Read Flow rate (US gallons/minute) for the final answer, computed by multiplying that velocity by the opening's cross-sectional area and converting to US GPM.
Worked example — 40 psi through a half-inch orifice
Start with a 40 psi pressure differential feeding a 0.5-inch orifice in water, density 1000 kg/m³ — the calculator's own defaults. Converting to SI first, 40 psi equals 275,790 Pa, so exit velocity is v = √(2 × 275,790 ⁄ 1000) = √551.58 = 23.486 m/s, the value shown in Exit velocity (m/s). That's the ideal jet speed leaving the hole, with essentially the entire pressure differential converted into motion.
The orifice's cross-sectional area is π × (0.0254 × 0.5 ⁄ 2)² = π × 0.00635² ≈ 0.0001267 m². Multiplying velocity by that area gives a volumetric flow of roughly 0.002975 m³/s, and scaling cubic meters per second up to US gallons per minute (× 15,850.32) lands on 47.16 GPM in the Flow rate (US gallons/minute) field. Because the model assumes a frictionless jet from a large reservoir, a real half-inch pipe fitting of that same length would likely deliver a bit less.
Questions
Why do I need pipe diameter and fluid density, not just pressure?
Because flow rate is a volume per time, and pressure alone is force per area — bridging the two requires knowing how large the opening is and how heavy the fluid is. Bernoulli's equation shows exit velocity depends only on pressure and density (v = √(2ΔP⁄ρ)), but velocity isn't flow: multiplying by the opening's cross-sectional area is what turns a speed into a volume rate, so diameter enters through plain geometry, not through the pressure physics itself.
How accurate is this compared to real-world plumbing?
It's an idealized, frictionless prediction, so it sits on the high side of what real hardware delivers. Actual orifices and pipe fittings lose energy to turbulence, viscosity, and wall roughness, and engineers apply a discharge coefficient — often around 0.6 for a sharp-edged orifice, up near 0.95 to 0.98 for a smoothly rounded nozzle — to bring the ideal number down to a measured one. Long or narrow pipe runs lose more to friction than a short, wide opening, so treat this figure as a ceiling, not a guarantee.
What does exit velocity actually mean here?
It's the speed of the fluid jet right as it leaves the orifice or pipe opening, derived from Torricelli's law — a special case of Bernoulli's equation for a tank draining under its own pressure. The model assumes the fluid surface inside the tank moves so slowly compared with the jet that its velocity can be treated as zero, which holds well when the tank's cross-section is much larger than the opening it's draining through.
Why is the default fluid density 1000 kg/m^3?
That's fresh water at roughly room temperature, the fluid this kind of calculation is most often run for — irrigation lines, fire hose nozzles, drain-down times, water-jet cutting setups. Swap in a different figure for another fluid: light hydrocarbons like gasoline sit around 720 to 770 kg/m³, seawater is close to 1025 kg/m³, and glycerin runs near 1260 kg/m³. Because velocity scales with the inverse square root of density, denser fluids exit slower for the same pressure push, and lighter ones exit faster.
Can I use this for a pump instead of a draining tank?
Not directly. This model is built for pressure-driven flow out of a large reservoir through a passive opening — no pump, no added mechanical energy. A pump's flow relates to head in a way that depends on that specific pump's own performance curve, not on this orifice equation, so pairing this calculator with a pump's discharge pressure would overstate the real flow unless the pump curve is brought in separately.
Does pipe length matter for this calculation?
Not in this idealized model — it treats the opening as a short orifice where friction along the walls is negligible. Real pipe runs longer than a few diameters lose additional pressure to wall friction, governed by the Darcy-Weisbach equation and the pipe's roughness, so a 40 psi differential pushed through fifty feet of narrow pipe delivers less flow than the same differential across a plain orifice plate. Longer, rougher runs pull the real answer further below this calculator's number.
Is converting PSI to GPM always this complicated?
For a real physical system, yes — flow rate is never a fixed multiple of pressure alone, which is why no single 'psi to gpm' factor appears anywhere in engineering references the way 2.54 appears for centimeters to inches. Manufacturers of specific nozzles, valves, or orifice plates do publish simplified charts, but those are just this same physics pre-solved for one fixed diameter and one fixed fluid — change either input and the chart stops applying.