Heat Transfer in Turbulent Flow

Why it matters

Heat transfer in turbulent flow is essential in many engineering applications, such as in the design of heat exchangers, cooling systems in power plants, and HVAC systems. Turbulent flow enhances the rate of heat transfer, making it crucial for efficient thermal management in various industrial processes.

Key ideas

• Turbulent Flow: Characterized by chaotic changes in pressure and flow velocity, turbulent flow occurs at high Reynolds numbers (typically greater than about 4000 for circular-pipe internal flow; this is not a universal external-flow threshold). It enhances mixing and increases the heat transfer rate compared to laminar flow.
• Reynolds Number (Re): A dimensionless number used to predict flow patterns in different fluid flow situations. It is calculated as Re = ρ·v·L/μ, where ρ is the fluid density, v is the velocity, L is the characteristic length, and μ is the dynamic viscosity.
• Nusselt Number (Nu): Represents the ratio of convective to conductive heat transfer across a boundary. For turbulent flow, empirical correlations are often used to calculate Nu, such as the Dittus-Boelter equation.
• Dittus-Boelter Equation: An empirical correlation for calculating the Nusselt number in turbulent flow, given by Nu = 0.023·Re^0.8·Pr^0.3, where Pr is the Prandtl number.
• Prandtl Number (Pr): A dimensionless number, Pr = ν/α, where ν is the kinematic viscosity and α is the thermal diffusivity. It indicates the relative thickness of the velocity and thermal boundary layers.

For fully developed turbulent flow in a smooth circular tube, Dittus–Boelter uses Nu_D = 0.023 Re_D^0.8 Pr^n, with n = 0.4 when heating the fluid and n = 0.3 when cooling it. Typical applicability is Re_D ≥ 10000, 0.7 ≤ Pr ≤ 160, adequate development length and moderate wall-to-bulk temperature differences. Evaluate fluid properties at the bulk mean temperature. See NPTEL correlation guidance.

Formulas

• Re = ρ·v·L/μ
  • ρ: Fluid density (kg/m³)
  • v: Velocity (m/s)
  • L: Characteristic length (m)
  • μ: Dynamic viscosity (Pa·s)
• Nu = 0.023·Re^0.8·Pr^0.3
  • Nu: Nusselt number (dimensionless)
  • Re: Reynolds number (dimensionless)
  • Pr: Prandtl number (dimensionless)
• Pr = ν/α
  • ν: Kinematic viscosity (m²/s)
  • α: Thermal diffusivity (m²/s)

Worked example

Given for cooling of fluid in a sufficiently long smooth circular tube, with fully developed flow and moderate property variation:

• Fluid density, ρ = 1000 kg/m³
• Velocity, v = 2 m/s
• Tube diameter, D = 0.05 m (use D as the characteristic length)
• Dynamic viscosity, μ = 0.001 Pa·s
• Prandtl number, Pr = 7

Steps:

1. Calculate the Reynolds number: Re = ρ·v·L/μ = 1000 kg/m³ · 2 m/s · 0.05 m / 0.001 Pa·s = 100,000
2. Use the Dittus-Boelter equation to find the Nusselt number: Nu = 0.023·Re^0.8·Pr^0.3 = 0.023·(100,000)^0.8·7^0.3 Nu ≈ 0.023·10000·1.79279 ≈ 412.34

Final Answer: Nu ≈ 412.34 (dimensionless)

Common mistakes

• Ignoring the flow regime: Not checking whether the flow is laminar or turbulent before applying formulas.
• Incorrect units: Mixing up units, especially in the Reynolds number calculation.
• Misapplying empirical correlations: Using the wrong correlation for the given flow conditions.

For GATE ME

Questions often involve calculating the Nusselt number or heat transfer coefficient in turbulent flow using empirical correlations. Practice problems on identifying flow regimes and applying the correct formulas based on given conditions.

Quick check

1. What is the typical Reynolds number range for turbulent flow?
2. Which empirical correlation is commonly used for calculating the Nusselt number in turbulent flow?
3. What does the Prandtl number represent?

Answers: 1. For internal circular-pipe flow, typically above about 4000; this alone does not establish the validity of a heat-transfer correlation; 2. Dittus-Boelter equation; 3. The relative thickness of the velocity and thermal boundary layers.