Flow Measurement

Why it matters

Flow measurement is crucial in various engineering applications, including water supply systems, chemical processing, and HVAC systems. Accurate flow measurement ensures efficient system operation, safety, and cost-effectiveness.

Key ideas

• Flow Measurement: The process of quantifying the bulk fluid movement. It can be measured in terms of volume flow rate or mass flow rate.
• Types of Flow Meters:
  • Orifice Meter: Uses a plate with a hole to create a pressure drop, measuring flow rate based on differential pressure.
  • Venturi Meter: A tube with a constricted throat that increases velocity and decreases pressure, used to measure flow rate.
  • Pitot Tube: Measures fluid flow velocity by comparing stagnation pressure with static pressure.
  • Rotameter: A variable area meter where a float rises in a tapered tube with increasing flow rate.
• Applications: Used in pipelines, open channels, and ducts to measure the flow of liquids and gases.

Formulas

• Continuity Equation: Q = A·v
  • Q: Volume flow rate (m³/s)
  • A: Cross-sectional area (m²)
  • v: Flow velocity (m/s)
• Bernoulli’s Equation: P₁ + 0.5·ρ·v₁² + ρ·g·h₁ = P₂ + 0.5·ρ·v₂² + ρ·g·h₂
  • P: Pressure (Pa)
  • ρ: Density (kg/m³)
  • v: Velocity (m/s)
  • g: Acceleration due to gravity (9.81 m/s²)
  • h: Height (m)

Worked example

Problem: Calculate the flow rate through a Venturi meter with an inlet diameter of 0.3 m and a throat diameter of 0.15 m. The pressure difference between the inlet and throat is 5000 Pa. Assume a horizontal meter, steady incompressible flow, uniform section velocities, negligible loss and water density 1000 kg/m³.

1. Calculate the cross-sectional areas:
   • A₁ = π/4 × (0.3)² = 0.0707 m²
   • A₂ = π/4 × (0.15)² = 0.0177 m²
2. Apply Bernoulli’s equation between inlet and throat:
   • P₁ + 0.5·ρ·v₁² = P₂ + 0.5·ρ·v₂²
   • 5000 + 0.5·1000·v₁² = 0.5·1000·v₂²
3. Use continuity equation:
   • A₁·v₁ = A₂·v₂
   • 0.0707·v₁ = 0.0177·v₂
4. Solve for v₁ and v₂:
   • v₂ = (A₁/A₂)·v₁ = 4·v₁
   • Substitute in Bernoulli’s equation: 5000 + 0.5·1000·v₁² = 0.5·1000·(4·v₁)²
   • 5000 + 500·v₁² = 8000·v₁²
   • 5000 = 7500·v₁²
   • v₁ = √(5000/7500) = 0.816497 m/s; v₂ = 3.26599 m/s
5. Calculate flow rate Q:
   • Q = (π/4)(0.3)²(0.816497) = 0.057715 m³/s

Answer: 0.057715 m³/s (ideal)

A real differential-pressure meter uses its calibrated discharge coefficient: Q = C_d A₂ sqrt[2Δp/(ρ(1-(A₂/A₁)²))] for the stated horizontal incompressible model. Gas flow may need an expansibility correction. A Pitot reading is local velocity, not automatically the pipe-average velocity.

Common mistakes

• Neglecting the effect of fluid density in Bernoulli’s equation.
• Incorrectly calculating the cross-sectional area.
• Forgetting to convert units where necessary.

For GATE ME

Questions often involve calculating flow rates using different types of flow meters, understanding the principles behind each type, and applying Bernoulli’s and continuity equations. Practice problems involving pressure differences and velocity calculations.

Quick check

1. What is the principle behind a Venturi meter?
2. How does a Pitot tube measure flow velocity?
3. What is the unit of volume flow rate?

Answers: 1. Pressure difference and continuity equation. 2. Uses the stagnation-minus-static pressure difference. 3. m³/s.