Bernoulli's Equation Calculator
Professional Bernoulli equation solver for engineers and students. Specify the unknown pressure, velocity, or elevation with optional head loss plus full SI/US unit handling.
Flow conditions
Supply known pressures, velocities, elevations, density, gravity, and head loss. Any single unknown variable can be solved from Bernoulli's energy balance.
Unit system
Choose gauge or absolute as long as both sides match.
Use the same reference as P1.
Non-negative values only; use 0 for negligible flow.
Must remain non-negative to keep the energy balance valid.
Any consistent datum; differences are what matter.
Match the datum used for z1.
Water: ~1000 kg/m³. Air: ~1.2 kg/m³.
Standard gravity is 9.80665 m/s² (32.174 ft/s²).
Major + minor losses; use 0 if negligible.
How to Use This Calculator
This professional-grade Bernoulli equation calculator helps engineers, students, and technical professionals analyze steady, incompressible flow between two points. Provide the known pressures, velocities, elevations, density, gravity, and any head loss. Pick the unknown variable from the dropdown and click Calculate to see the solved value plus converted outputs.
Values update automatically with debounced input changes, and the solver enforces a consistent unit system internally to avoid rounding drift.
Methodology
The tool assembles Bernoulli's energy equation between Point 1 and Point 2: P1/(ρg) + v1²/(2g) + z1 = P2/(ρg) + v2²/(2g) + z2 + hL. The selected unknown is isolated algebraically, so the equilibrium always balances the sum of pressure, velocity, gravitational, and head loss contributions.
Once the unknown is determined, the engine recomputes both sides of the energy balance to report the resulting energy heads and pressure difference. All intermediate calculations run in double precision and convert back to the units you selected for display.
Data Source and Limits
Authoritative Source: Munson, Young and Okiishi’s Fundamentals of Fluid Mechanics, 8th Edition (2016), Wiley. Calculations follow the textbook relations for steady, incompressible, inviscid flow with optional head loss.
The calculator assumes consistent units across both points. Input pressure references (absolute or gauge) must match. Use the head loss field to bundle major and minor losses computed separately.
Glossary
- P1, P2 — Pressure at point 1 and 2 (Pa, kPa, MPa, bar, or psi)
- v1, v2 — Average velocity at point 1 and 2 (m/s or ft/s)
- z1, z2 — Elevation head at point 1 and 2 (m or ft)
- ρ — Fluid density (kg/m³ or slug/ft³)
- g — Gravitational acceleration (m/s² or ft/s²)
- hL — Head loss between points (m or ft)
- Heads — Pressure head P/(ρg), velocity head v²/(2g), elevation head z
Worked example
Given water (ρ = 1000 kg/m³), g = 9.80665 m/s², P1 = 200 kPa, v1 = 2 m/s, z1 = 2 m, z2 = 0 m, v2 = 5 m/s, and hL = 1 m. Solve for P2.
Use: P2 = P1 + 0.5ρ(v1² − v2²) + ρg(z1 − z2 − hL)
Compute terms: 0.5·ρ·(v1² − v2²) = 0.5·1000·(4 − 25) = −10500 Pa; ρg(z1 − z2 − hL) = 1000·9.80665·1 = 9806.65 Pa.
Therefore, P2 ≈ 199306.65 Pa (≈ 199.31 kPa).
Frequently Asked Questions
When is Bernoulli applicable?
Steady, incompressible, inviscid flow along a streamline with no machinery between the points. Losses can be added through hL.
How do I include frictional losses?
Estimate hL using Darcy–Weisbach and minor loss coefficients, then enter that combined head loss here.
Are the results gauge or absolute?
They honor whatever reference you used for both P1 and P2; only the difference matters.
What if the computed velocity is not real?
A negative radicand (inside the square root) means the supplied pressures, elevations, or head loss are inconsistent—double-check each entry.
Does the calculator support US units?
Yes. Switch the unit system to set imperial defaults, or choose units per field. Behind the scenes everything converts to base SI for reliable math.
Can I solve for upstream variables too?
Yes. Pick P1, v1, or z1 in the Solve for dropdown to have the calculator compute the upstream unknown instead.
What precision is used?
Double precision internally, with results rounded to practical significant digits for display.
Full original guide (expanded)
The original deployment included an audit spine with formulas, variable definitions, and validation notes. The engine now surfaces those validated formulas via the dedicated Formulas detail and the assurance badges below.
The audit noted: "Audit: Complete" along with a verification date of 2026-01-19. Verified sources were CalcDomain's Engineering and Mechanical collections plus Wiley's Fundamentals of Fluid Mechanics.