Hong Life CalculatorsUnderstanding the dimensionless numbers calculator — concept and principle
Heat-transfer engineers describe problems with dimensionless groups so that results from one size, fluid or speed can be applied to another. This calculator evaluates four of the most common: the Nusselt, Prandtl, Biot and Fourier numbers, from the heat transfer coefficient, a characteristic length and basic material properties.
How it is calculated
- Nusselt: Nu = h·L / k_fluid, convective heat transfer relative to pure conduction across the fluid.
- Prandtl: Pr = c_p·μ / k, momentum diffusivity relative to thermal diffusivity of the fluid.
- Biot: Bi = h·L / k_solid, internal conduction resistance of a solid relative to its surface convection resistance.
- Fourier: Fo = α·t / L², dimensionless time for transient conduction (α in m²/s, t in s, L in m).
Nu and Bi have the same form but use different conductivities, so the calculator has separate inputs: k for the fluid (Nu) and k_s for the solid (Bi).
The principle
Each group is a ratio of two competing effects. A large Nu means the flow carries heat far better than still fluid would; Nu = 1 corresponds to pure conduction. Pr compares how quickly velocity and temperature disturbances spread, so it sets the relative thickness of the velocity and thermal boundary layers. Bi tells you whether a solid heats uniformly (small Bi) or develops internal gradients (large Bi). The Grashof number, Gr = gβΔT·L³/ν², which drives natural convection, needs expansion coefficient and kinematic viscosity inputs and is not computed here.
Worked example
- Inputs: h = 50 W/m²K, L = 0.05 m, k = 0.6 W/m·K, c_p = 4180 J/kg·K, μ = 0.001 Pa·s, α = 1.4 × 10⁻⁷ m²/s, t = 60 s (properties close to water near room temperature).
- Nu = 50 × 0.05 / 0.6 ≈ 4.167.
- Pr = 4180 × 0.001 / 0.6 ≈ 6.967.
- Bi = 50 × 0.05 / 0.6 ≈ 4.167 (the default k_s is also 0.6, so Bi equals Nu here).
- Fo = 1.4 × 10⁻⁷ × 60 / 0.05² = 0.00336.
| Fluid conductivity k_f (W/m·K) | Nusselt Nu = hL/k |
|---|---|
| 0.15 | 16.66666667 |
| 0.3 | 8.333333333 |
| 0.45 | 5.555555556 |
| 0.6 | 4.166666667 |
| 0.75 | 3.333333333 |
| 0.9 | 2.777777778 |
| 1.2 | 2.083333333 |
How to read the result
Typical Pr values: about 0.7 for air, about 7 for water at 20 °C, and hundreds or more for oils. If Bi < 0.1, a solid can be treated as having uniform temperature (lumped capacitance). A small Fo, as here, means heat has penetrated only a thin outer layer; one-term series solutions for transient conduction are usually accurate once Fo exceeds about 0.2.
Common mistakes
- Leaving k_s at the default 0.6 instead of the solid's conductivity (steel about 50, aluminium about 200 W/m·K).
- Choosing an inconsistent characteristic length.
- Mixing dynamic viscosity (Pa·s) with kinematic viscosity (m²/s).
Frequently asked questions
What characteristic length should I use?
It depends on the correlation: plate length for flat plates, diameter for tubes and cylinders, and V/A for lumped transient analysis.
How do I get h from Nu?
Rearrange: h = Nu·k / L, with Nu taken from a suitable correlation such as Dittus–Boelter for turbulent pipe flow.
Sources: Incropera, DeWitt, Bergman and Lavine, Fundamentals of Heat and Mass Transfer, Wiley · Bird, Stewart and Lightfoot, Transport Phenomena, Wiley