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.
Understanding Engineering: Biot and Nusselt Numbers
A useful way to think about the Biot number is to imagine heat taking two possible routes. Heat must move from the middle of a solid toward its surface. It must then cross the surface into the surrounding fluid.
If the solid is a poor conductor, or if the object is thick, the trip inside the solid is slow. The centre can stay much hotter or colder than the surface for a long time. This is common in baking a potato, cooling a large metal casting, or heating a plastic part.
If the solid conducts heat very well and is small, its interior catches up quickly. A small metal bead in moving water is much closer to this case.
The choice of characteristic length matters because it represents the distance heat must travel inside the object. For many shapes, volume divided by exposed area gives a sensible length. This value becomes smaller when an object has lots of surface for its size.
Thin wires, small chips, and metal foils therefore tend to change temperature more evenly than thick blocks made from the same material. Students should not assume that every dimension is the correct length. A sphere, cylinder, plate, and fin each have geometry that affects the heat path.
The exposed area must include only surfaces actually touching the fluid. A face pressed against insulation does not provide convection to the air.
The convection coefficient links both numbers to real conditions at the surface. It is not a fixed property of air or water. It changes with fluid speed, surface shape, orientation, temperature difference, and whether the flow is smooth or turbulent.
Still air removes heat slowly because a warm layer of air can remain near the surface. A fan disrupts that layer and brings in cooler air.
Water usually transfers heat more strongly than air because it conducts heat better and can carry more thermal energy. This explains why a computer heat sink works far better with airflow and why a hot object cools quickly when placed in flowing water.
The Nusselt number is often found from an experimental correlation rather than calculated from first principles. Engineers use results measured for cases such as air flowing over a flat plate, water inside a pipe, or natural convection around a vertical wall. These correlations usually depend on other dimensionless groups.
Reynolds number describes the balance between fluid inertia and viscosity. Prandtl number compares momentum spreading with heat spreading in a fluid. For buoyancy driven flow, Grashof number measures the importance of density changes caused by temperature differences.
When using a correlation, pay close attention to its stated geometry, flow range, and boundary condition. A correlation for turbulent pipe flow cannot safely be used for quiet air around a box. The numbers are powerful because they organize many physical effects, but they only give reliable answers when the physical situation matches the model.
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.