Bernoulli Equation Calculator

Whether you're routing pipes, sizing a nozzle, or just checking how flow changes across a constriction, it boils down to how pressure, velocity, and elevation interact between two points. The Bernoulli Equation captures those relationships, letting you solve for one unknown if you know the rest — pressure, velocity, height, and density at the points of interest. You'll see this principle in practical use for piping layouts, aerodynamics, and duct design. Below you'll find the core formula, an example, a deeper dive into the theory, and a FAQ that should clear up most typical questions.

What is the Bernoulli Equation?

Bernoulli’s equation basically says that, for a flowing incompressible fluid with minimal losses, the sum of its static pressure, kinetic energy, and potential energy stays the same between two points along the flow. Lose some in one term, and you'll see it pop up in another.

Simple Explanation

If you’ve ever pinched a garden hose, you’ve noticed the water shoots out faster but the pressure drops near the exit. That’s Bernoulli’s principle at work — increase speed, and pressure falls. Add height, and pressure and speed have to readjust. It's just a fixed pot of energy being shared between those three terms.

📐 Browse all 1000+ Interactive Calculators

Bernoulli Flow System Diagram

Bernoulli Equation Calculator

How to Use This Calculator

1. Enter values for pressure (P), velocity (v), and height (h) at Point 1 and Point 2 — leave exactly 1 field blank for the unknown you want to solve.
2. Enter the fluid density (ρ) — use 1000 kg/m³ for water or 1.225 kg/m³ for air at sea level.
3. Select your unit system: Metric (Pa, m/s, m) or Imperial (psi, ft/s, ft).
4. Click Calculate to see your result.

📹 Video Walkthrough — How to Use This Calculator

Bernoulli Equation Formulas

To get pressure, velocity, or height at any point, this is the go-to equation.

Simple Example

There’s water in a horizontal pipe (so h₁ = h₂ = 0). Point 1 has P₁ = 200,000 Pa, v₁ = 2 m/s. At Point 2, speed increases to v₂ = 4 m/s. Use water density, ρ = 1000 kg/m³.

P₂ = P₁ + ½ρ(v₁² − v₂²) = 200,000 + ½(1000)(4 − 16) = 200,000 − 6,000 = 194,000 Pa

So, if you double the flow speed, the pressure drops by 6,000 Pa in this setup.

Technical Analysis: Understanding Bernoulli's Principle

Whenever you’re dealing with fluids in motion, pressure and velocity are linked. Bernoulli's equation is the standard tool for relating those, plus any elevation changes, between two points. It expresses the conservation of energy for steady, incompressible, low-viscosity flows — a starting point for many practical engineering jobs.

Fundamental Physics Behind Bernoulli's Equation

Bernoulli’s equation follows straight from conservation of energy: if you track a chunk of fluid moving along a line (streamline) and the flow is steady, then the sum of its pressure head, velocity head, and elevation head won’t change. The three terms — pressure (P), kinetic (½ρv²), and gravitational potential (ρgh) — represent different ways a fluid carries energy.

The pressure term (P) is force per unit area, originating from molecular collisions inside the fluid. The velocity term (½ρv²) is just the usual kinetic energy, scaled for fluid, per unit volume. ρgh comes from the change in height, or what you’d get from lifting or dropping the fluid against gravity.

Keep in mind, these forms only hold when the following are pretty much true: no major viscosity losses, fluid density doesn’t noticeably change, you’re comparing two points along the same streamline, and nobody’s adding or removing energy with a pump or a turbine between your chosen points.

Practical Applications in Engineering Systems

Bernoulli’s principle shows up in plenty of places. In aerospace, it’s used to estimate lift on wings: faster airflow on top means lower static pressure, which helps explain how lift is generated (though real wings are more complex). Civil engineers use it for things like sizing municipal water lines or figuring how fast water comes out of a dam’s outlet.

For automation and actuator systems, especially pneumatic or hydraulic versions, being able to estimate velocities and resulting pressures using Bernoulli’s equation is useful for checking if your system’s components are sized in the right ballpark.

In HVAC, the relationship between pressure drop and velocity lets you size ductwork and choose fans to balance performance with energy efficiency and noise. You can quickly check if planned changes in duct diameter or route cause velocities (and thus noise and loss) to get too high.

Worked Example: Pipe Flow Analysis

Suppose you have a horizontal pipe running water from a large tank through a narrower section. Point 1: diameter 0.2 m, v₁ = 2 m/s, P₁ = 150,000 Pa. Point 2: diameter drops to 0.1 m. What's the pressure at Point 2?

First, continuity law: A₁v₁ = A₂v₂

π(0.1)²(2) = π(0.05)²(v₂)

v₂ = 2 × (0.1/0.05)² = 8 m/s

Pipe is horizontal (h terms drop out), so Bernoulli gives:

150,000 + ½(1000)(2²) = P₂ + ½(1000)(8²)

150,000 + 2,000 = P₂ + 32,000

P₂ = 120,000 Pa

When the pipe narrows, speed jumps and pressure falls — classic Bernoulli effect, and a quick check of what happens in a real fluid system.

Design Considerations and Limitations

Bernoulli’s equation is often a good estimate, but real fluids aren’t ideal. Viscosity means some energy becomes heat or eddies, and isn’t available for pressure or velocity. At higher velocities (for gases, over Mach 0.3), density starts to change, so incompressible flow formulas don’t quite work anymore. If the flow’s changing with time or you’re between a pump and an outlet, the actual energy budget will be different.

To correct for this, engineers add a loss term (hL) for friction and turbulence. The revised formula is: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ + hL, with hL coming from charts or empirical correlations based on pipe roughness, bends, valves, and Reynolds number.

Watch out for density and temperature changes, especially for air or other gases — typical values may be fine for rough work, but precision demands correct properties at actual conditions. This becomes important if you’re running actuators or valves in hot or cold environments or at altitude.

Advanced Applications and Modern Engineering

Modern CFD models are built on Bernoulli’s concepts, then expanded with all the messy real-world details: turbulence, heat, compression, and multi-phase flows. For simple cases, Bernoulli gets you most of the way; for anything complex, CFD or empirical testing becomes necessary to get accurate predictions.

Even in new fields — wind turbines, hydro turbines, microfluidics — Bernoulli’s principle is behind the pressure and velocity tradeoffs, whether you’re maximizing output or controlling flow in a tiny channel. The scale or application changes, but the energy bookkeeping doesn’t.

When designing automation or fluid systems for tough or unusual conditions, being able to quickly estimate pressure drops or velocity changes lets you set expectations and spot problems early, long before the analysis needs to get fancy.

Integration with Other Fluid Mechanics Principles

In practice, you rarely use Bernoulli alone. Mass conservation (continuity) gives you velocities, while momentum considerations (forces and reactions) are why pipe elbows need bracing or why jets kick back. Pump sizing, valve drops, and network balancing all come from layering these equations together.

If you’re deep into system design, chances are you’ll also use friction loss equations, pump or fan curves, and real flow testing alongside Bernoulli’s equation for a full solution. The calculator here is a shortcut for the core relationships, not the final say on a whole fluid system.

For further calculations or more specialized tools, check out our engineering calculators section for system-specific calculators and more ways to cross-check your designs.

📐 Browse all 1000+ Interactive Calculators →