Library / Heat Transfer / HTR-10
Heat Transfer
Biot number
Compares surface convection with internal conduction.
- Ref
- HTR-10
- Rev
- v1
- Field
- Heat Transfer
- Result
- Bi
h is the surface heat transfer coefficient, L the characteristic length of the solid (volume divided by surface area is the usual choice) and k the thermal conductivity of the solid, not of the fluid. Bi is the ratio of the resistance to heat flow inside the body to the resistance at its surface.
Compares surface convection with internal conduction. Below about 0.1 the object stays almost uniform in temperature and lumped analysis is valid.
Calculator
- Characteristic length at which Bi reaches 0.10.05 m
- Largest h for which Bi stays below 0.1500 W/m²·K
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Bi = 0.02 at the defaults: a 10 mm steel part in a strong air flow stays within 2 % of uniform in temperature, so it can be treated as a single lump.
- If Bi is above 0.1, the surface cools well before the centre does, and the full conduction equation is needed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Thermal conductivity | 0.1 to 500 W/m·K | 0.002 | 10 | 9.998 | +10% | −9.09 % |
| h Heat transfer coefficient | 1 to 10,000 W/m²·K | 2e-4 | 2 | 2 | +10% | +10 % |
| L Characteristic length | 1e-4 to 1 m | 2e-4 | 2 | 2 | +10% | +10 % |
Sweep of Heat transfer coefficient
| Heat transfer coefficient (W/m²·K) | Biot number |
|---|---|
| 1 | 2e-4 |
| 3.162 | 6.325e-4 |
| 10 | 0.002 |
| 31.62 | 0.006325 |
| 100 | 0.02 |
| 316.2 | 0.06325 |
| 1,000 | 0.2 |
| 3,162 | 0.6325 |
| 10,000 | 2 |
Sweep one variable
Variables and how each one contributes
Biot number
What the equation gives you. Everything else feeds it.
Heat transfer coefficient
The convection coefficient between the surface and the fluid, in W/m²·K. Directly proportional: a stronger flow or a liquid raises Bi and makes the inside of the body lag behind the surface.
Characteristic length
The characteristic length of the solid, V/A, in m. Directly proportional: a thicker body takes longer for its inside to follow the surface.
Thermal conductivity
The thermal conductivity of the solid, in W/m·K. Inverse: a good conductor (copper, 400) keeps its temperature uniform; a poor one (steel, 15, or plastic) does not.
Worked example with the values above
- Numerator: h L = 100 × 0.01 = 1 W/m²·K·m.
- Divide by the conductivity of the solid: 1 / 50.
- Bi = 0.02, below 0.1, so the lumped analysis is valid.
Assumptions and limits
- Steady convection at the surface with a constant h.
- A solid whose conductivity is the same everywhere and whose shape is compact (no long thin sections).
Common mistakes
- Using the conductivity of the fluid. That gives the Nusselt number instead.
- Using the half-thickness, the radius or V/A inconsistently when comparing against the 0.1 limit.
Statistics and uncertainty
Bi is a plain product of ratios, so errors add as relative errors. Because the usual threshold of 0.1 is a rule of thumb (it keeps the lumped model within about 5 %), an uncertainty of 30 % in h, common for convection, is acceptable unless Bi is already near 0.1.
Where it is used
- Deciding whether a part can be treated as one temperature when it heats or cools.
- Checking the range of validity of lumped cooling.
Maths you need
- Thermal conductivity: k
- Convection coefficient: h