Bernoulli Theorem Calculator

P1 equals rho g times (P2 over rho g plus V2 squared minus V1 squared over 2g plus Z2 minus Z1 plus h)

Solution

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How It Works

Bernoulli's equation balances pressure energy, kinetic energy, and potential energy between two points along a streamline in a steady, incompressible flow. When one form of energy increases, the others decrease to keep the total constant. The head loss term captures real-world friction and turbulence losses that the ideal equation ignores.

Example Problem

Water (density 1,000 kg/m³) flows through a horizontal pipe. At point 1 the pressure is 200,000 Pa and velocity is 2 m/s. At point 2 the velocity is 4 m/s. Both elevations are equal and head loss is zero. What is the pressure at point 2?

  1. Since Z₁ = Z₂ and h = 0, the equation simplifies to: P₁ + ½ρV₁² = P₂ + ½ρV₂²
  2. 200,000 + 0.5 × 1,000 × 4 = P₂ + 0.5 × 1,000 × 16
  3. P₂ = 202,000 − 8,000 = 194,000 Pa

Frequently Asked Questions

What is Bernoulli's equation used for?

It relates pressure, velocity, and elevation at two points in a fluid flow. Engineers use it to size pipes, design Venturi meters, predict pressure drops, and analyze siphons and nozzles.

Does Bernoulli's equation work for gases?

It works for low-speed gas flows where density changes are negligible (Mach < 0.3). For high-speed compressible flows, use the compressible energy equation instead.

How much head loss is typical in a pipe system?

Head loss varies widely. A 100 m run of 50 mm steel pipe carrying water at 2 m/s may lose 5–15 m of head depending on roughness and fittings. Use the Darcy-Weisbach equation for precise estimates.

What assumptions does Bernoulli's equation make?

The equation assumes steady, incompressible flow along a single streamline with no energy added by pumps. Viscous losses are handled separately through the head loss term.

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