Bernoulli Equation Calculator

About the Bernoulli Equation Calculator

Bernoulli's equation is one of the most important relationships in fluid mechanics, relating pressure, velocity, and height at two points along a streamline in steady, incompressible, inviscid flow. It is the mathematical expression of energy conservation for flowing fluids.

This calculator solves Bernoulli's equation for any of the three unknowns at point 2: velocity, pressure, or height. Enter the conditions at point 1 (pressure, velocity, height) along with the known values at point 2, select what to solve for, and get the result with dynamic pressure, hydrostatic pressure, and total head.

Presets for common scenarios — garden hose nozzles, venturi meters, water towers, airplane wings, and fire hoses — demonstrate how the equation applies across engineering and everyday physics. An energy balance bar visualizes the relative contributions of static pressure, dynamic pressure, and hydrostatic pressure. It keeps the input and output states visible together so the pressure-velocity tradeoff is easier to follow, especially when you are comparing nozzle sizing or elevation changes.

When This Page Helps

Bernoulli's equation is essential for pipe sizing, nozzle design, venturi flow meters, and aerodynamics. This calculator eliminates the tedious algebra of rearranging the equation for different unknowns and provides immediate unit conversions.

The energy balance visualization helps students intuitively understand how static pressure, dynamic pressure, and elevation head trade off along a streamline. That makes the calculator useful for both classroom examples and first-pass design checks.

How to Use the Inputs

1. Select which variable to solve for at point 2 (velocity, pressure, or height).
2. Choose a fluid from the dropdown or enter a custom density.
3. Enter pressure, velocity, and height at point 1.
4. Enter the known values at point 2.
5. Read the calculated result and energy breakdown from the outputs.
6. Use the energy balance bar to visualize pressure contributions.

Example Calculation

Tips & Best Practices

• Total pressure (static + dynamic + hydrostatic) is conserved along a streamline.
• Narrowing a pipe increases velocity and decreases pressure (venturi effect).
• This calculator uses 1 atm = 101,325 Pa.
• For air at sea level, density ≈ 1.225 kg/m³.
• For pipe design, also account for friction losses (not included in ideal Bernoulli).

Applications of Bernoulli's Principle

Bernoulli's equation is applied in venturi flow meters (measuring flow rate from pressure drop), pitot tubes (measuring aircraft airspeed), carburetors (mixing fuel and air), perfume atomizers, and water distribution systems. The Torricelli theorem for fluid draining from a tank is a direct consequence of Bernoulli's equation with the top surface at atmospheric pressure.

Limitations and Extensions

The ideal Bernoulli equation assumes no friction, no heat transfer, and incompressible flow. In reality, engineers use extended versions: the modified Bernoulli equation adds a friction loss term, and for compressible flows (Mach > 0.3), the compressible Bernoulli or isentropic flow equations must be used instead.

Historical Context

Daniel Bernoulli published this principle in his book Hydrodynamica in 1738, though the mathematical formulation commonly used today was developed later by Leonhard Euler. The equation represents one of the earliest applications of energy conservation to fluid mechanics and remains one of the most widely used equations in engineering.

Frequently Asked Questions

• What is Bernoulli's equation?

  It states that the total mechanical energy per unit volume of a flowing fluid remains constant along a streamline: P + ½ρv² + ρgh = constant. When velocity increases, pressure decreases, and vice versa.

• When is Bernoulli's equation valid?

  It applies to steady, incompressible, inviscid (frictionless) flow along a streamline. It is a good approximation for low-speed liquid flow with minimal friction losses.

• Why does pressure drop when velocity increases?

  Energy conservation: if fluid speeds up (higher kinetic energy), it must lose pressure energy to keep the total constant. This is the venturi effect.

• How does this explain airplane lift?

  Air flowing faster over the curved top of a wing has lower pressure than the slower air below. The pressure difference creates an upward net force (lift). Bernoulli's principle partially explains this.

• What about friction losses in real pipes?

  Real pipe flows have friction losses. Engineers add a head-loss term to the equation (Darcy-Weisbach equation) to account for viscous dissipation in long pipes.

• Can I use this for gas/air flow?

  Yes, for low-speed air flow (below Mach 0.3). Above Mach 0.3, compressibility effects become significant and the incompressible Bernoulli equation loses accuracy.