Calculator Inputs
Enter convection, conductivity, and a characteristic length from geometry. The calculator converts supported units to SI before evaluating Bi.
Formula Used
The Biot number compares internal conduction resistance to external convection resistance:
Bi = (h · Lc) / k
- h is the convection coefficient (W/m²·K).
- k is the material conductivity (W/m·K).
- Lc is the characteristic length, commonly Lc = V/As.
Characteristic Length Options
- Plane wall of thickness t: Lc = t/2.
- Long cylinder of radius r: Lc = r/2.
- Sphere of radius r: Lc = r/3.
- Custom: enter Lc directly for specialized shapes.
How to Use This Calculator
- Enter h and choose its unit.
- Enter k and choose its unit.
- Select a geometry to determine Lc.
- Provide thickness, radius, or custom Lc with units.
- Click Calculate to see Bi and an interpretation.
- Use CSV or PDF buttons to export the results.
Example Data Table
These sample scenarios show how geometry and properties shift Biot number magnitude.
| Case | h (W/m²·K) | k (W/m·K) | Geometry | Size input | Computed Lc (m) | Bi | Typical takeaway |
|---|---|---|---|---|---|---|---|
| Polymer slab in air | 15 | 0.2 | Wall | t = 0.02 m | 0.0100 | 0.7500 | Internal gradients are important |
| Metal sphere in air | 25 | 200 | Sphere | r = 0.05 m | 0.0167 | 0.0021 | Lumped analysis is often reasonable |
| Foam cylinder with forced flow | 80 | 0.04 | Cylinder | r = 0.01 m | 0.0050 | 10.0000 | Large internal resistance dominates |
Biot Number in Thermal Design
The Biot number (Bi) compares internal conduction resistance to external convection resistance. It tells you whether a solid stays nearly isothermal during heating or cooling, or whether strong internal gradients develop that demand a conduction model. This decision directly affects accuracy, runtime, and measurement needs for safer design decisions in practice.
Core Definition and Units
Bi = h·Lc/k, where h is the convection coefficient (W/m²·K), k is the solid conductivity (W/m·K), and Lc is a characteristic length (m). Bi is dimensionless, so it scales cleanly across sizes and materials.
Choosing the Characteristic Length
A robust choice is Lc = V/A (volume divided by exposed area). It reflects how “thick” a body is relative to the area exchanging heat and works well when multiple faces lose heat. Common results: plane wall of thickness 2L uses Lc≈L; long cylinder uses Lc≈r/2; sphere uses Lc≈r/3.
Convection Coefficient: Realistic Ranges
h depends on flow and fluid. Natural convection in air is often 2–10 W/m²·K. Forced air commonly ranges 10–100 W/m²·K. Water convection can span 100–10,000 W/m²·K. Radiation can be handled with an equivalent h when temperatures and emissivity are known.
Material Conductivity Sets the Trend
Large k (metals) drives Bi downward, encouraging uniform internal temperature. Small k (polymers, wood, foams, many ceramics) raises Bi for the same geometry and h, making internal gradients more likely and slowing heat penetration.
Interpreting Biot Thresholds
A common design rule is Bi<0.1 for the lumped-capacitance assumption, where the solid can be treated as one temperature. For Bi≈0.1–1, gradients are noticeable and simplified transient solutions may work. For Bi>1, spatially resolved conduction is usually required.
Connection to Transient Cooling and Heating
Bi pairs with the Fourier number Fo = αt/Lc² in transient analysis. Fo measures how much conduction has progressed in time, while Bi indicates how strongly the surface couples to the fluid. Standard charts and series solutions are organized by Bi and Fo.
Practical Workflow for Engineers
Compute Bi early: pick Lc from geometry, estimate h from correlations or experience, and use k at the operating temperature. Small Bi supports quick energy-balance modeling and simpler experiments. Large Bi suggests gradients, thermal stress risk, and a need for detailed boundary conditions and property data.
Frequently Asked Questions
What does a low Biot number mean?
A low Bi means internal conduction is fast compared with surface convection, so the solid’s temperature stays nearly uniform. This supports the lumped-capacitance approach and simplifies transient heating or cooling calculations.
When is lumped capacitance acceptable?
A common rule is Bi < 0.1 using an appropriate characteristic length. If this holds and properties are roughly constant, you can model the object with a single temperature that changes over time.
How do I choose the characteristic length Lc?
Use Lc = V/A for many real parts because it accounts for exposed area. For classic shapes, common choices are L (half-thickness) for a wall, r/2 for a long cylinder, and r/3 for a sphere.
Does Biot number depend on time?
Not directly. Bi uses geometry, h, and k, so it is fixed for a given setup. However, h and k can vary with temperature or flow, which can make Bi change during operation.
What if I have convection and radiation together?
Combine them by using an effective surface coefficient: h_eff = h_conv + h_rad. Then compute Bi with h_eff. This is useful when radiation is significant compared with convection.
Why can Bi be large for insulating materials?
Insulators have low thermal conductivity k, which increases h·Lc/k. Even with modest convection, thick foam or plastic parts can develop strong internal temperature gradients, so a spatial conduction model is often needed.
Can Biot number help with thermal stress risk?
Yes. Higher Bi implies larger internal gradients during transients, which can drive differential expansion and stress. It’s an early screening tool before you invest in detailed finite-difference or finite-element simulations.