Bernoulli Equation

Bernoulli's equation expresses energy conservation along a streamline: pressure, velocity and elevation heads trade off. Learn heads, Venturi and orifice meters, Pitot tubes, Torricelli's law and the extended energy equation with losses and pumps.

Where fluid speeds up, its pressure drops. That surprising fact explains how carburettors draw fuel, how Venturi meters measure flow, how Pitot tubes give an aircraft its airspeed, and why roofs lift off in storms.

Bernoulli's equation — with loss and pump terms added — is the working equation for sizing every pumped piping system.

An analogy

A roller coaster: height (potential energy) converts to speed (kinetic energy) and back. In a pipe, pressure also joins the exchange.

Energy in a flowing fluid

Each kilogram of fluid carries pressure energy, kinetic energy and potential energy. With no friction or machines, their sum stays constant along a streamline. Expressed as heads (metres of fluid), the terms are easy to compare.

Assumptions

Steady, incompressible, inviscid flow along a streamline, with no pumps/turbines between points. Real systems add head loss h_L (friction, fittings) and pump head h_p or turbine head h_t.

Flow measurement

A Venturi narrows the flow; the pressure drop reveals velocity and flow rate (C_d ≈ 0.98). Orifice plates are cheaper but lose more energy (C_d ≈ 0.61). Pitot–static tubes measure local velocity from stagnation minus static pressure — aircraft airspeed indicators work this way.

Energy and hydraulic grade lines

The EGL plots total head along a system and the HGL plots p/ρg + z. The EGL drops with losses and jumps at pumps; where the HGL falls below the pipe, pressure is sub-atmospheric (risk of air ingress or cavitation). For compressible flow, the energy equation uses enthalpy, and Bernoulli in its simple form applies only below Mach ≈ 0.3.

Formulas

• Bernoulli (ideal): p/ρg + V²/2g + z = constant
• Energy equation (real): p₁/ρg + V₁²/2g + z₁ + h_pump = p₂/ρg + V₂²/2g + z₂ + h_L
• Venturi / orifice: Q = C_d A₂ √(2Δp / (ρ(1 − β⁴))), β = d/D
• Torricelli: V = √(2 g h)
• Pitot tube: V = √(2(p₀ − p) / ρ)

Worked example: Venturi meter

Given: D = 100 mm, d = 50 mm (β = 0.5); Δp = 20 kPa, water; C_d = 0.98

Find: Flow rate.

• A₂ = 1.963 × 10⁻³ m²
• √(2 × 20 000/(998 × 0.9375)) = 6.54 m/s
• Q = 0.98 × 1.963 × 10⁻³ × 6.54 = 0.0126 m³/s

Answer: ≈ 12.6 L/s

Worked example: Draining a tank

Given: Water 5 m above an outlet

Find: Jet velocity.

• V = √(2 × 9.81 × 5) = 9.9 m/s

Answer: 9.9 m/s

Common mistakes

• Wrong: Applying ideal Bernoulli across a pump or long pipe. — Right: Use the energy equation with h_pump and h_L.
• Wrong: Mixing gauge and absolute pressures between points. — Right: Use the same basis at both points.

Applications

• Flow metering: Venturi, orifice and nozzle meters (ISO 5167) in water, steam and gas lines.
• Aviation: Pitot–static systems give airspeed; wing lift is often explained partly via pressure differences.
• Siphons & ejectors: Low-pressure zones draw fluid in carburettors, spray guns and steam ejectors.

Key takeaways

• p/ρg + V²/2g + z = constant (ideal).
• Add pump head and losses for real systems.
• Venturi, orifice and Pitot measure flow via Δp.
• Low pressure risks cavitation — check HGL.

Going further

• Kinetic-energy correction: Non-uniform profiles need α ≈ 2 (laminar) or ≈ 1.05 (turbulent) on the V²/2g term.
• Unsteady Bernoulli: Adds ∫∂V/∂t ds — needed for water hammer and oscillating flows.
• Compressible Bernoulli: For gases at speed, isentropic relations replace the incompressible form (stagnation pressure grows with Mach number).