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Bernoulli Equation Calculator
Solve P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ for any one unknown. Tick continuity to link the velocities through the pipe diameters (A₁v₁ = A₂v₂), which also lets you find the flow from a Venturi pressure difference.
Last updated:
Solve for
kg/m³
Point 1
m/s
m
Point 2
m/s
m
Enter your values to see the result.
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How to use the Bernoulli Equation Calculator
- Choose the unknown to solve for.
- Pick the fluid and the pressure unit.
- Tick continuity and enter both diameters if the points are in one pipe of changing size.
- Enter the known pressures, velocities and heights at points 1 and 2.
- Read the result, flow rate and energy breakdown.
Bernoulli and continuity equations
| Equation | Form |
|---|---|
| Bernoulli (energy per volume) | P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ |
| Bernoulli (heads, metres) | P/ρg + v²/2g + h = constant |
| Continuity | A₁v₁ = A₂v₂, so v₂ = v₁ (D₁/D₂)² |
| Venturi velocity | v₁ = √[2(P₁ − P₂) / (ρ((D₁/D₂)⁴ − 1))] |
| Torricelli | v = √(2gh) |
| Dynamic pressure | q = ½ρv² |
Worked example
Water (998 kg/m³) flows at 2 m/s and 200 kPa in a 100 mm pipe, which narrows to 50 mm and rises 1 m. Continuity gives v₂ = 2 × (100/50)² = 8 m/s. Then P₂ = 200,000 + ½ × 998 × (2² − 8²) + 998 × 9.80665 × (0 − 1) = 200,000 − 29,940 − 9,787 ≈ 160.3 kPa.
Dynamic pressure at common speeds
| Speed | Water (998 kg/m³) | Air (1.225 kg/m³) |
|---|---|---|
| 1 m/s | 0.50 kPa | 0.61 Pa |
| 5 m/s | 12.5 kPa | 15.3 Pa |
| 10 m/s | 49.9 kPa | 61.3 Pa |
| 30 m/s | 449 kPa | 551 Pa |
| 100 m/s | — | 6.13 kPa |
Frequently asked questions
What is the Bernoulli equation?
For steady, incompressible, frictionless flow along a streamline, P + ½ρv² + ρgh is constant. P is static pressure, ½ρv² the dynamic pressure and ρgh the hydrostatic term. Faster flow at the same height means lower pressure.
How do I use continuity with Bernoulli?
In a closed pipe the volume flow is the same everywhere: A₁v₁ = A₂v₂, so v₂ = v₁ (D₁/D₂)². Halving the diameter makes the fluid four times faster, and Bernoulli then tells you how much the pressure drops.
How does a Venturi meter measure flow?
It measures the pressure drop between the full pipe and a narrow throat. Combining both equations gives v₁ = √(2ΔP / (ρ(k² − 1))) with k = (D₁/D₂)², and the flow is Q = A₁v₁. Choose Solve for v₁ with continuity ticked.
What is Torricelli's law?
Water leaving a small hole a height h below the free surface of an open tank flows out at v = √(2gh), which is Bernoulli with equal (atmospheric) pressures and zero surface velocity. For a 3 m head that is about 7.7 m/s.
Should I use gauge or absolute pressure?
Either, as long as both points use the same kind. Only the difference matters in the equation. A negative result in gauge pressure just means below atmospheric.
When is Bernoulli's equation not accurate?
It ignores friction losses, pumps, turbines and compressibility. For long pipes add head loss (Darcy–Weisbach), and for gases faster than about Mach 0.3 use compressible-flow equations.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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