Heat Transfer in Laminar Flow

Why it matters

Understanding heat transfer in laminar flow is crucial for designing efficient thermal systems, such as heat exchangers and cooling systems in electronics. It helps engineers predict how heat will move through fluids in controlled environments, ensuring optimal performance and safety.

Key ideas

• Laminar Flow: Occurs when a fluid flows in parallel layers, with no disruption between them. It is characterized by low Reynolds numbers (for ordinary circular-pipe flow, roughly below 2300).
• Heat Transfer Mechanisms: In laminar flow, heat transfer occurs primarily through conduction within the fluid layers and convection at the boundary layers.
• Thermal Boundary Layer: The region of fluid near a heated surface where temperature gradients exist. Its thickness affects the rate of heat transfer.
• Nusselt Number (Nu): A dimensionless number representing the ratio of convective to conductive heat transfer across the boundary layer.
• Prandtl Number (Pr): A dimensionless number that relates the fluid's momentum diffusivity to its thermal diffusivity.

Formulas

• Re = ρ·V·D / μ

  • Re: Reynolds number (dimensionless)
  • ρ: Density of fluid (kg/m³)
  • V: Velocity of fluid (m/s)
  • D: Characteristic length (m)
  • μ: Dynamic viscosity (Pa·s)

• Nu = h·L / k

  • Nu: Nusselt number (dimensionless)
  • h: Convective heat transfer coefficient (W/m²·K)
  • L: Characteristic length (m)
  • k: Thermal conductivity of fluid (W/m·K)

• Pr = μ·c_p / k

  • Pr: Prandtl number (dimensionless)
  • μ: Dynamic viscosity (Pa·s)
  • c_p: Specific heat capacity at constant pressure (J/kg·K)
  • k: Thermal conductivity of fluid (W/m·K)

Worked example

Given:

• Fluid: Water
• Mean velocity (V): 0.02 m/s
• Diameter of pipe (D): 0.05 m
• Density (ρ): 1000 kg/m³
• Dynamic viscosity (μ): 0.001 Pa·s
• Thermal conductivity (k): 0.6 W/m·K
• Specific heat capacity (c_p): 4186 J/kg·K

Steps:

1. Calculate the Reynolds number: Re = ρ·V·D / μ = 1000 kg/m³ · 0.02 m/s · 0.05 m / 0.001 Pa·s = 1000

   • Re = 1000 is in the laminar regime. Assume hydrodynamically and thermally fully developed flow in a circular pipe, constant wall temperature, constant properties, negligible axial conduction and viscous dissipation.

2. Calculate the Prandtl number: Pr = μ·c_p / k = 0.001 Pa·s · 4186 J/kg·K / 0.6 W/m·K = 6.977

3. Assume a Nusselt number for laminar flow in a pipe (Nu = 3.66 for fully developed laminar flow with constant wall temperature): Nu = 3.66

4. Calculate the convective heat transfer coefficient: h = Nu·k / D = 3.66 · 0.6 W/m·K / 0.05 m = 43.92 W/m²·K

Final Answer: 43.92 W/m²·K

For fully developed laminar circular-pipe flow with uniform wall heat flux instead, Nu ≈ 4.36. Entrance regions and other duct shapes require different relations. The same constant Nu does not apply to every laminar flow.

Common mistakes

• Misidentifying the flow regime (laminar vs. turbulent) based on incorrect Reynolds number calculations.
• Using incorrect characteristic lengths for calculating dimensionless numbers.
• Neglecting the effects of thermal boundary layers in heat transfer calculations.

For GATE ME

Questions often involve calculating dimensionless numbers like Reynolds, Prandtl, and Nusselt numbers, and using them to find heat transfer coefficients. Practice problems involving different fluids and flow conditions to strengthen understanding.

Quick check

1. What is the typical Reynolds number range for laminar flow?
2. How does the Nusselt number relate to heat transfer?
3. What does the Prandtl number signify in heat transfer?

Answers: 1. Approximately Re_D < 2300 for ordinary circular-pipe flow, with transition dependent on disturbances; 2. It represents the ratio of convective to conductive heat transfer; 3. It relates momentum diffusivity to thermal diffusivity.