A venturi meter used to measure flow speed in the pipe. Derive an expression for the flow speed "H1" interns of the crossectional areas "A1" and "A2" and the difference in height "h" of the liquid levels in the two vertical tubes ?

v₁ = √[ 2gh / ((A₁ / A₂)² − 1) ]

Explanation:

Use Bernoulli's equation:

P₁ + ½ ρ v₁² + ρgz₁ = P₂ + ½ ρ v₂² + ρgz₂

Since there's no elevation change between points 1 and 2, z₁ = z₂.

P₁ + ½ ρ v₁² = P₂ + ½ ρ v₂²

Assuming incompressible fluid, the volumetric flow rate is the same at points 1 and 2.

Q₁ = Q₂

v₁ A₁ = v₂ A₂

v₂ = v₁ A₁ / A₂

Substituting:

P₁ + ½ ρ v₁² = P₂ + ½ ρ (v₁ A₁ / A₂)²

P₁ + ½ ρ v₁² = P₂ + ½ ρ v₁² (A₁ / A₂)²

P₁ − P₂ = ½ ρ v₁² (A₁ / A₂)² − ½ ρ v₁²

P₁ − P₂ = ½ ρ v₁² ((A₁ / A₂)² − 1)

v₁² = 2 (P₁ − P₂) / (ρ ((A₁ / A₂)² − 1))

v₁² = 2 (ρgh) / (ρ ((A₁ / A₂)² − 1))

v₁² = 2gh / ((A₁ / A₂)² − 1)

v₁ = √[ 2gh / ((A₁ / A₂)² − 1) ]

where are the vectors ?