Bernoulli Equation
The Bernoulli equation describes a steady flow of an incompressible fluid. According to the Bernoulli principle, the total pressure of the fluid (static + dynamic + hydrostatic) remains constant along a streamline:
p + ½ρv² + ρgh = constant
where:
- p — static pressure at the chosen point (Pa or psi)
- ρ (rho) — fluid density (kg/m³ or slug/ft³); constant for incompressible fluids
- v — flow speed at the chosen point (m/s or ft/s)
- h — elevation of the chosen point (m or ft)
- g — acceleration due to gravity: 9.80665 m/s²
Comparing Two Points Along a Streamline
Applying Bernoulli's equation to two points on the same streamline:
p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂
If you know five of the six values (p₁, v₁, h₁, p₂, v₂, h₂), you can solve for the sixth using this calculator. Select "Solve for" from the dropdown and enter the remaining five values.
Example
Fluid: water (ρ = 1000 kg/m³), h₁ = h₂ = 3 m (same elevation), p₁ = 1000 Pa, v₁ = 2 m/s, p₂ = 1200 Pa. What is v₂?
1000 + ½·1000·4 + 1000·3·9.80665 = 1200 + ½·1000·v₂² + 1000·3·9.80665
3000 = 1200 + 500·v₂²
v₂ = √3.6 ≈ 1.897 m/s
Flow Rate
For a pipe with known cross-sectional area A, the volumetric flow rate is:
Q = A × v
For incompressible flow, continuity requires A₁v₁ = A₂v₂. Enable the optional Pipe Area input to calculate Q at both points.
Unit Systems
| Quantity | Metric (SI) | American |
|---|---|---|
| Pressure | Pa | psi |
| Velocity | m/s | ft/s |
| Elevation | m | ft |
| Density | kg/m³ | slug/ft³ |
| Flow rate | m³/s | ft³/s |
| Head | m | ft |
FAQs
What does "incompressible fluid" mean?
An incompressible fluid has constant density throughout the flow (e.g., water, oil). Gases such as air can be treated as incompressible at low Mach numbers (Ma < 0.3). For compressible flows at higher speeds, the full compressible Bernoulli equation is needed.
What are the "heads"?
The Bernoulli equation is often expressed as a sum of three heads (all in units of length):
- Pressure head = p / (ρg)
- Velocity head = v² / (2g)
- Elevation head = h
- Total head = pressure head + velocity head + elevation head = constant
When does pressure decrease as velocity increases?
When elevation is constant (h₁ = h₂), Bernoulli's equation reduces to p + ½ρv² = constant. If velocity increases (e.g., fluid passes through a narrower section), static pressure must decrease. This is the principle behind the Venturi meter and aircraft lift.
What fluid presets are available?
- Water: 1000 kg/m³ (1.940 slug/ft³)
- Seawater: 1025 kg/m³ (1.987 slug/ft³)
- Air: 1.225 kg/m³ (0.002377 slug/ft³) at standard conditions
- Oil: 870 kg/m³ (1.686 slug/ft³) typical light crude
- Mercury: 13600 kg/m³ (26.35 slug/ft³)
- Custom: enter any density value