The Biot number and Nusselt number are dimensionless tools engineers use to compare heat transfer effects without depending on one specific object size or fluid. The Biot number tells whether temperature inside a solid stays nearly uniform or develops strong internal gradients during heating or cooling. The Nusselt number tells how much convection at a surface improves heat transfer compared with pure conduction through a still fluid layer.
Together, they help engineers design cooling systems, heat exchangers, electronics packages, and thermal treatments of materials.
The Biot number compares internal conduction resistance inside a solid to convection resistance at its surface, using Bi = hLc/k_s. A small Biot number, usually Bi < 0.1, means the solid can often be modeled as having one uniform temperature, called the lumped-capacitance approximation. The Nusselt number compares convective heat transfer to conductive heat transfer in the fluid, using Nu = hL/k_f.
A larger Nusselt number usually means stronger fluid motion, thinner thermal boundary layers, and greater heat transfer from the surface.
Key Facts
- Biot number: Bi = hLc/k_s, where h is convection coefficient, Lc is characteristic length, and k_s is solid thermal conductivity.
- Nusselt number: Nu = hL/k_f, where L is a chosen length scale and k_f is fluid thermal conductivity.
- Lumped-capacitance model is usually valid when Bi < 0.1.
- Characteristic length for a solid in Bi is often Lc = V/A_s, where V is volume and A_s is exposed surface area.
- For lumped cooling or heating: (T - T_infinity)/(T_i - T_infinity) = e^(-hA_s t/(rho V c_p)).
- Nu = 1 represents pure conduction across a fluid layer in the simplest reference case, while Nu > 1 indicates convection enhancement.
Vocabulary
- Biot number
- A dimensionless number that compares heat conduction resistance inside a solid with convection resistance at its surface.
- Nusselt number
- A dimensionless number that measures how much convection enhances heat transfer compared with conduction through the fluid.
- Convection coefficient
- The constant h that relates surface heat flux to the temperature difference between a surface and the surrounding fluid.
- Characteristic length
- A representative length scale used in dimensionless heat transfer formulas, often V/A_s for Biot number calculations.
- Lumped-capacitance approximation
- A thermal model that treats a solid as having a uniform temperature throughout its volume at any instant.
Common Mistakes to Avoid
- Using the fluid thermal conductivity in the Biot number, which is wrong because Bi compares conduction inside the solid to convection at the surface and must use k_s.
- Using the solid thermal conductivity in the Nusselt number, which is wrong because Nu describes heat transfer through the fluid boundary layer and must use k_f.
- Assuming Bi < 0.1 means heat transfer is small, which is wrong because it only means internal temperature gradients in the solid are small.
- Using diameter, radius, or thickness without checking the required characteristic length, which can give the wrong Bi or Nu because the length scale depends on geometry and correlation.
Practice Questions
- 1 A steel sphere has radius 0.02 m, k_s = 45 W/(m K), and is cooled by air with h = 60 W/(m^2 K). For a sphere, Lc = V/A_s = r/3. Calculate Bi and decide whether lumped capacitance is reasonable.
- 2 Water flows over a heated flat plate with h = 800 W/(m^2 K), L = 0.50 m, and k_f = 0.60 W/(m K). Calculate the Nusselt number.
- 3 Two identical hot metal cylinders are cooled in different fluids. Case A has a low h and Bi = 0.03, while Case B has a high h and Bi = 0.8. Explain which case is more likely to have strong internal temperature gradients and why.