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Bernoulli Equation Calculator

Bernoulli's equation governs how pressure, velocity, and elevation change as a fluid moves through a system. Engineers use it to design pumping systems, predict nozzle performance, and size pipe networks. This calculator solves the complete form across two points on a streamline, letting you find missing pressure, speed, or height values, and compute volumetric and mass flow rates from pipe diameter and velocity data.

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Creators Dominik Czernia, PhD
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Bernoulli's Equation for Two Points

When an incompressible fluid flows steadily along a streamline, the total energy per unit volume remains constant between any two locations. This relationship allows you to solve for unknown pressures, velocities, or elevations if you know enough other parameters.

p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂

Δp = p₂ − p₁

Q = π(d/2)² × v

ṁ = Q × ρ

  • p₁, p₂ — Static pressure at positions 1 and 2
  • ρ — Fluid density (constant for incompressible flow)
  • v₁, v₂ — Fluid velocity at positions 1 and 2
  • h₁, h₂ — Elevation head at positions 1 and 2
  • g — Gravitational acceleration (9.81 m/s²)
  • Q — Volumetric flow rate
  • ṁ — Mass flow rate
  • d — Pipe diameter

Understanding the Three Energy Components

Bernoulli's principle states that three forms of pressure energy exist along a streamline:

  • Static pressure (p): The ordinary pressure exerted by the fluid on pipe walls. At a faucet, you measure this with a gauge.
  • Dynamic pressure (½ρv²): Energy from the fluid's motion. Faster-moving fluid has more dynamic pressure but lower static pressure.
  • Hydrostatic pressure (ρgh): The weight of fluid above a point. Water at 10 m depth feels 98 kPa more pressure than at the surface.

These three sum to a constant along any streamline in steady, incompressible flow. If velocity increases through a narrower pipe section, pressure must drop to conserve energy—a principle that governs everything from carburetors to garden hose nozzles.

Flow Rate Calculations from Pipe Diameter

Once you know the fluid velocity and pipe diameter, calculating volumetric flow rate is straightforward. The cross-sectional area of the pipe is π(d/2)², so multiplying by velocity gives the volume passing through per unit time:

  • Volumetric flow remains constant along the streamline in incompressible flow (continuity equation). If pipe diameter shrinks, velocity must increase proportionally.
  • Mass flow is volumetric flow multiplied by fluid density. For water at ~1000 kg/m³, a 50 L/min flow equals about 0.83 kg/s.
  • Practical example: A garden hose (19 mm diameter) carrying water at 2 m/s delivers roughly 570 L/hour; attach a 6 mm nozzle and velocity jumps to 20 m/s but flow volume stays constant.

When Bernoulli's Equation Applies

Bernoulli's equation is valid only under four strict conditions:

  • Incompressible fluid: Density is constant. Water, oil, and other liquids qualify; gases at low speed (~0.3 Mach) also work, but not sonic flows.
  • Inviscid (frictionless) flow: Ideal for short sections and rough estimates, but real pipes lose energy to friction. Use extended versions like the Darcy–Weisbach equation for long runs.
  • Steady flow: Conditions at each point don't change with time. Pumps, open channels, and gravity-fed systems meet this; pulsing or surging flows do not.
  • Along a streamline: You must compare two locations on the same flow path. Crossing between different streamlines invalidates the equation.

Violation of these assumptions—such as analyzing a compressible gas at high Mach number—requires modified forms and is beyond this basic tool's scope.

Common Pitfalls and Real-World Considerations

Misapplying Bernoulli's equation or ignoring friction losses are frequent sources of design errors.

  1. Neglecting Friction Losses — Bernoulli assumes a frictionless fluid, but real pipes, fittings, and valves dissipate energy. Over long distances (50+ meters) or high Reynolds numbers, friction drop can equal or exceed the Bernoulli prediction. Always apply a friction factor correction or empirical loss coefficients in engineering designs.
  2. Confusing Gauge and Absolute Pressure — Pressure gauges show gauge pressure (pressure above atmospheric). Bernoulli's equation requires absolute pressure. If a pump outlet reads 2 bar gauge and atmospheric is 1.01 bar, use 3.01 bar absolute in the equation, or your height and velocity predictions will be off by ~100 kPa.
  3. Assuming Constant Density for Gases — Air and other gases compress noticeably under pressure changes above ~0.3 Mach. If your gas flow involves pressure swings greater than 20–30% or speeds above 100 m/s, density will vary significantly and Bernoulli's incompressible form breaks down. Use compressible flow equations instead.
  4. Ignoring Elevation in Horizontal Pipes — In a long horizontal pipe, elevation terms (ρgh) are identical at both points, so they cancel. However, in vertical risers, a 10 m height difference adds ~100 kPa of pressure drop (for water). Overlooking this is a leading cause of pump-sizing errors in multi-story buildings.

Frequently Asked Questions

What does Bernoulli's equation tell you about velocity and pressure?

Bernoulli's equation reveals an inverse relationship: when a fluid accelerates and dynamic pressure rises, static pressure falls. Think of air flowing over an airplane wing: the curved upper surface forces air to travel faster, creating a low-pressure zone that produces lift. Conversely, in a diffuser section where flow slows down, pressure increases. This trade-off is central to countless industrial and natural phenomena, from carburetors to blood flow in narrowed arteries.

Why is flow rate constant along a streamline?

The continuity equation—a consequence of mass conservation—dictates that if an incompressible fluid enters a pipe section, the same volume must exit each second. When diameter shrinks, velocity must increase to maintain constant volumetric flow. For example, water exiting a 25 mm hose through a 5 mm nozzle tip accelerates roughly 25 times. This is why pinching a hose makes water spray farther: you are trading large volume at low speed for small volume at high speed.

Can you use Bernoulli's equation for gases?

Yes, but with limits. At low speeds (below ~100 m/s or Mach 0.3) and modest pressure changes, gases behave nearly incompressibly. Standard air at room temperature under these conditions works fine. However, at high speeds or large pressure ratios—such as in supersonic flows or compressors—gas density changes significantly and you must switch to compressible flow equations. Ignoring compressibility in high-speed applications introduces large errors.

What is the most common real-world application of Bernoulli's principle?

Pumping and distribution systems rely heavily on Bernoulli's equation. Engineers use it to predict pressure drops, size pipes, and ensure adequate flow to taps and sprinklers. Another familiar example is the venturi effect: narrowing a pipe to measure or accelerate flow. Medical devices like nebulisers and insecticide sprayers exploit this principle. Even in nature, Bernoulli explains how blood vessels maintain flow when they constrict, and how spinning soccer balls curve through the air.

How do you account for energy loss due to friction?

Bernoulli's equation assumes zero friction, but real systems lose energy to viscous drag, turbulence, and fitting resistance. The Darcy–Weisbach equation adds a friction term: Δp_friction = f × (L/D) × (½ρv²), where f is the friction factor, L is pipe length, and D is diameter. For rough estimates, empirical loss coefficients (K values) for elbows, valves, and tees sum up quickly. In long pipelines or at high Reynolds numbers (>10,000), friction losses often dominate and must not be ignored.

What happens when you measure pressure with a pitot tube?

A pitot tube stagnates the flow at its tip, converting all dynamic pressure to a pressure rise. The difference between the stagnation point (tip) and a static port elsewhere on the tube gives you dynamic pressure: ½ρv². Rearranging to solve for v gives airspeed or water velocity. Aircraft and flow measurement devices use pitot tubes because they are simple, inexpensive, and reliable for moderate speeds. At very high speeds or in compressible flow, compressibility corrections become necessary.

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