Discharge Through a Venturi Meter

Equations: Bernoulli between sections 1 and 2 of a horizontal, ideal meter: p₁/ρ + V₁²/2 = p₂/ρ + V₂²/2, so V₂² − V₁² = 2Δp/ρ. Continuity: Q = A₁V₁ = A₂V₂, so V₁ = V₂(A₂/A₁). Combining them: V₂ = √[(2Δp/ρ) / (1 − (A₂/A₁)²)] and Q = A₂V₂. Values: Δp = 30 kPa = 30,000 Pa is the pressure difference read between the two taps. ρ = 1000 kg/m³ is the density of water. Area goes with diameter squared, so A₂/A₁ = (D₂/D₁)² = (0.10/0.20)² = 0.25. The throat area is A₂ = π(0.10)²/4 = 0.00785 m². Step 1: 2Δp/ρ = 2(30,000)/1000 = 60 m²/s². Step 2: 1 − (A₂/A₁)² = 1 − 0.25² = 0.9375. Step 3: V₂ = √(60/0.9375) = √64 = 8.0 m/s. Step 4: Q = A₂V₂ = 0.00785 × 8.0 = 0.0628 m³/s. Using the diameter ratio 0.5 instead of the area ratio 0.25 gives V₂ = 8.94 m/s. Leaving Δp in kPa instead of Pa gives a velocity about 30 times too small.