Bernoulli Equation Calculator

Solve Bernoulli's equation for the pressure at a second point along a streamline, given fluid density and the pressure, velocity, and elevation at two points.

Quick Facts

Formula
P + ½ρv² + ρgh = constant
Static, dynamic, and hydrostatic pressure sum to a constant along a streamline for steady, incompressible, frictionless flow.

Your Results

Calculated
Pressure at Point 2
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P₂ solved from Bernoulli's equation
Dynamic pressure change
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½ρ(v₁² − v₂²)
Hydrostatic pressure change
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ρg(h₁ − h₂)
Bernoulli constant
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Total pressure, same at both points

Ready

Enter fluid density plus pressure, velocity, and elevation at Point 1, and velocity and elevation at Point 2, then press Calculate.

Understanding the Bernoulli Equation

The Bernoulli equation is a statement of energy conservation for a fluid moving along a streamline. For steady, incompressible, frictionless (inviscid) flow, the sum of three pressure terms — static, dynamic, and hydrostatic — stays constant from one point to another, even as the fluid speeds up, slows down, or changes elevation. It explains why fluid pressure drops where it speeds up through a constriction, why an airfoil generates lift, and how a Venturi meter or Pitot tube can measure flow rate from a pressure reading.

The formula

For two points, 1 and 2, along the same streamline:

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

  • P is the static (absolute) pressure at the point, in pascals.
  • ρ (rho) is the fluid density, in kg/m³ — about 1000 for water and 1.2 for air at sea level.
  • v is the flow velocity at the point, in m/s. The term ½ρv² is the dynamic pressure.
  • g is standard gravity, 9.80665 m/s², and h is elevation, in meters. The term ρgh is the hydrostatic pressure.

How this calculator works

Enter the fluid density and the known pressure, velocity, and elevation at Point 1, plus the velocity and elevation at Point 2. The calculator rearranges the equation above to solve for the unknown pressure at Point 2:

P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂)

It also reports the dynamic pressure change (the effect of the velocity difference alone), the hydrostatic pressure change (the effect of the elevation difference alone), and the Bernoulli constant — the total pressure at Point 1, which should equal the total pressure at Point 2 as a built-in check that energy is conserved between the two points.

Assumptions and limits

This form of the equation assumes steady flow (conditions don't change with time), an incompressible fluid (constant density — a good approximation for liquids and for gases well below the speed of sound), no viscous friction losses along the path, and no pump or turbine adding or removing energy between the two points. Real pipe flow includes friction, which the Darcy-Weisbach or Hazen-Williams equations account for separately as head loss.

Frequently Asked Questions

What is Bernoulli's equation?
Bernoulli's equation states that for steady, incompressible, frictionless flow along a streamline, the sum of static pressure, dynamic pressure, and hydrostatic pressure is constant: P + ½ρv² + ρgh = constant. This calculator uses it to solve for the pressure at a second point given the fluid density and the pressure, velocity, and elevation at a first point, plus the velocity and elevation at the second point.
What units should I use?
Enter fluid density in kilograms per cubic meter (1000 for water, about 1.2 for air), pressure in kilopascals, velocity in meters per second, and elevation in meters. The calculator converts pressure to pascals internally and uses standard gravity (9.80665 m/s²) for the hydrostatic term.
When does Bernoulli's equation not apply?
The equation assumes steady, incompressible, frictionless (inviscid) flow with no pump or turbine adding or removing energy between the two points. It breaks down with significant friction losses in long pipes, turbulent or compressible high-speed flow, and any unsteady (time-varying) flow — use head-loss equations like Darcy-Weisbach, or full CFD, for those cases.

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