Bernoulli's equation and the venturimeter

Bernoulli's equation is conservation of energy for a moving fluid. Along a streamline, for steady, incompressible flow without friction, total head stays constant.

The equation

pρ g + V²2 g + z = constant

• p/ρg is the pressure head
• V²/2g is the velocity head
• z is the datum (elevation) head

Each term is in metres of fluid. Between two sections of a real pipe we add the head loss h_L:

p₁ρ g + V₁²2 g + z₁ = p₂ρ g + V₂²2 g + z₂ + h_L

Venturimeter

A venturimeter narrows the pipe to a throat. Velocity rises at the throat, so pressure drops. Measuring that drop with a manometer gives the discharge.

With continuity (A₁V₁ = A₂V₂) and Bernoulli between inlet and throat:

Q = C_d A₁ A₂√(A₁² − A₂²) √(2 g h)

h is the difference in piezometric head, and C_d is about 0.96 to 0.98.

With a mercury manometer reading x across water:

h = x ( S_mS − 1 ) = 12.6 x

Worked example

A 200 mm × 100 mm venturimeter carries water, and a mercury manometer reads 150 mm. Take C_d = 0.98.

h = 0.15 × 12.6 = 1.89 m of water

A₁ = 0.0314 m² and A₂ = 0.00785 m², so

Q = 0.98 × 0.0314 × 0.00785√(0.0314² − 0.00785²) × √(2 × 9.81 × 1.89) = 0.0484 m³/s

That is about 48.4 litres per second.

Other uses

The same equation explains the Pitot tube (velocity from stagnation pressure), flow from an orifice (V = √(2gh), Torricelli) and why siphons have a height limit.