# Bernoulli Equation Solver (Velocity / Pressure / Elevation)

Solve the Bernoulli equation p/(ρg) + v²/(2g) + z = const for steady, ideal, incompressible flow between two points on a streamline. Supply all three upstream quantities (p₁, v₁, z₁) and any two of the three downstream quantities; the tool solves for the third. Solve velocity: v₂ = √[v₁² + 2(p₁-p₂)/ρ + 2g(z₁-z₂)] — a negative radicand is physically impossible and is rejected. Solve pressure: p₂ = p₁ + ½ρ(v₁²-v₂²) + ρg(z₁-z₂). Solve elevation: z₂ = z₁ + (p₁-p₂)/(ρg) + (v₁²-v₂²)/(2g). g = 9.81 m/s². Density in kg/m³/g/cm³, pressure in Pa/kPa/bar/atm/psi, velocity in m/s, elevation in m/cm/ft. The upstream pressure/velocity/elevation heads and the total head are also reported.

> Canonical page: https://elysiatools.com/en/tools/bernoulli-equation-solver

- **Category:** Math & Numbers

- **Keywords:** Bernoulli, Bernoulli equation, streamline, pressure head, velocity head, elevation head, total head, fluid mechanics, ideal flow, incompressible

## Overview

The Bernoulli Equation Solver calculates downstream velocity, pressure, or elevation for steady, ideal, incompressible flow along a streamline. Enter the three upstream values, two downstream values, fluid density, and units to obtain the missing quantity along with upstream pressure, velocity, elevation, and total heads.

## Inputs

- **Unknown to Solve** (select): Which downstream quantity to solve for: velocity v₂, pressure p₂, or elevation z₂. Provide the other two downstream quantities.
- **Fluid Density ρ** (number): Fluid density ρ. Enter in the selected Density Unit.
- **Density Unit** (select)
- **Upstream Pressure p₁** (number): Upstream (point 1) pressure p₁. Enter in the selected Pressure Unit.
- **Upstream Velocity v₁** (number): Upstream (point 1) velocity v₁ in m/s.
- **Upstream Elevation z₁** (number): Upstream (point 1) elevation z₁. Enter in the selected Length Unit.
- **Pressure Unit** (select)
- **Velocity Unit** (select)
- **Length Unit** (select)
- **Downstream Pressure p₂ (leave blank if solving p₂)** (number): Downstream (point 2) pressure p₂. Enter in the selected Pressure Unit. Required unless solving for p₂.
- **Downstream Velocity v₂ (leave blank if solving v₂)** (number): Downstream (point 2) velocity v₂ in m/s. Required unless solving for v₂.
- **Downstream Elevation z₂ (leave blank if solving z₂)** (number): Downstream (point 2) elevation z₂. Enter in the selected Length Unit. Required unless solving for z₂.
- **Decimal Places** (number)

## When to use

- Calculate downstream velocity from pressure and elevation changes in an ideal flow.
- Find downstream pressure when velocity and elevation at both points are known.
- Determine the downstream elevation required to maintain the same Bernoulli head.

## How it works

- Select whether to solve for downstream velocity v₂, pressure p₂, or elevation z₂.
- Enter upstream pressure p₁, velocity v₁, elevation z₁, and fluid density.
- Enter the two downstream quantities that are known, leaving the selected unknown blank.
- Choose supported units and decimal places, then review the calculated result and reported upstream heads.

## Use cases

- Estimate velocity changes through a pressure-reducing or Venturi-style section.
- Check pressure and elevation relationships in ideal pipeline or channel-flow calculations.
- Verify fluid mechanics homework, design calculations, and streamline energy balances.

## Frequently asked questions

### What equation does the tool solve?

It solves p/(ρg) + v²/(2g) + z = constant using g = 9.81 m/s².

### Which downstream value can I calculate?

You can calculate downstream velocity v₂, pressure p₂, or elevation z₂.

### What values are required?

Provide p₁, v₁, z₁, fluid density, and two of p₂, v₂, and z₂.

### Which units are supported?

Density supports kg/m³ and g/cm³; pressure supports Pa, kPa, bar, atm, and psi; elevation supports m, cm, and ft; velocity uses m/s.

### What happens if the velocity calculation has a negative radicand?

The tool rejects it because a negative radicand is physically impossible for the velocity formula.

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