Internal Flow

Why it matters

Internal flow is essential in engineering applications such as HVAC systems, water supply networks, and chemical processing. Understanding internal flow helps in designing efficient systems for transporting fluids through pipes and ducts.

Key ideas

• Laminar and Turbulent Flow: Internal flow can be classified as laminar or turbulent based on the Reynolds number. Laminar flow is smooth and orderly, while turbulent flow is chaotic.
• Reynolds Number: A dimensionless number used to predict flow patterns in different fluid flow situations. It is calculated as Re = ρ·v·D / μ, where ρ is the fluid density, v is the velocity, D is the characteristic length (e.g., diameter of a pipe), and μ is the dynamic viscosity.
• Pressure Drop: As fluid flows through a pipe, it experiences a loss of pressure due to friction and other factors. This is quantified using the Darcy-Weisbach equation.
• Darcy-Weisbach Equation: Used to calculate the pressure drop in a pipe, given by ΔP = f·(L/D)·(ρ·v²/2), where f is the friction factor, L is the length of the pipe, and ΔP is the pressure drop.
• Friction Factor: Depends on the flow regime and the roughness of the pipe's interior surface. For laminar flow, f = 64/Re, and for turbulent flow, it is determined using the Moody chart or empirical correlations.

Conditions and friction convention

Use mean axial velocity and the Darcy friction factor f_D. The Fanning factor is f_D/4. For fully developed laminar flow of a Newtonian fluid in a circular pipe, f_D = 64/Re. The usual approximate ranges are laminar below Re ≈ 2300, transitional between about 2300 and 4000, and turbulent above; disturbances can shift transition. Noncircular ducts use hydraulic diameter for many correlations but do not generally retain the circular-pipe laminar coefficient 64.

The friction expression gives the dissipative pressure-loss term. It equals the actual static-pressure drop in a horizontal constant-area pipe without machinery; elevation and velocity changes require the energy equation. Add local losses separately.

Formulas

• Re = ρ·v·D / μ

  • Re: Reynolds number (dimensionless)
  • ρ: Fluid density (kg/m³)
  • v: Fluid velocity (m/s)
  • D: Characteristic length (m)
  • μ: Dynamic viscosity (Pa·s)

• ΔP = f·(L/D)·(ρ·v²/2)

  • ΔP: Pressure drop (Pa)
  • f: Friction factor (dimensionless)
  • L: Length of the pipe (m)
  • D: Diameter of the pipe (m)
  • ρ: Fluid density (kg/m³)
  • v: Fluid velocity (m/s)

Worked example

Given: A water pipe with a diameter of 0.1 m, length of 50 m, water density of 1000 kg/m³, dynamic viscosity of 0.001 Pa·s, and velocity of 2 m/s.

1. Calculate the Reynolds number: Re = ρ·v·D / μ = 1000·2·0.1 / 0.001 = 200000

   • The flow is turbulent since Re > 4000.

2. Use a supplied Darcy friction factor f = 0.02 for this exercise. Roughness is not specified, so it cannot be independently selected from a Moody chart.

3. Calculate the pressure drop using the Darcy-Weisbach equation: ΔP = f·(L/D)·(ρ·v²/2) = 0.02·(50/0.1)·(1000·2²/2) ΔP = 0.02·500·2000 = 20000 Pa

Final Answer: 20000 Pa

Common mistakes

• Confusing the units of viscosity and density.
• Incorrectly identifying the flow regime (laminar vs. turbulent).
• Using the wrong friction factor for the flow regime.

For GATE ME

Questions often involve calculating the Reynolds number, pressure drop, or friction factor. Practice problems on identifying flow regimes and using the Darcy-Weisbach equation.

Quick check

1. What is the Reynolds number for a fluid with ρ = 850 kg/m³, v = 1.5 m/s, D = 0.05 m, and μ = 0.002 Pa·s?
2. What is the pressure drop in a 10 m pipe with f = 0.03, D = 0.1 m, ρ = 1000 kg/m³, and v = 3 m/s?
3. What flow regime is indicated by a Reynolds number of 1500?

Answers: 1. 31875 2. 13500 Pa 3. Laminar