Grashof and Rayleigh Numbers
Theory: how strong buoyancy-driven flow is
Overview
Natural-convection correlations use either the Grashof or the Rayleigh number, which confuses me. What's the difference?
The Grashof number tells how strong thermal buoyancy is relative to viscosity — how readily the fluid starts moving. The Rayleigh number multiplies it by the Prandtl number, adding how well the moving fluid carries heat, giving the strength of natural convection. For heat transfer you normally use Rayleigh. The difference shows clearly when comparing fluids with very different Prandtl numbers, like air and oil.
Definitions
$\beta$ is the volumetric expansion coefficient (the reciprocal of absolute temperature for an ideal gas), $\Delta T$ the temperature difference, $L$ a characteristic length (height for a vertical plate), $\nu$ kinematic viscosity and $\alpha$ thermal diffusivity. The cube of length means larger objects have stronger natural convection and turn turbulent sooner.
Natural or forced?
The Richardson number compares buoyancy with inertia. As a guide, above 10 natural convection dominates, below 0.1 forced convection dominates, and in between is mixed convection. Whether the flow aids or opposes buoyancy also changes heat transfer.
The two people behind the names
The Grashof number honours Franz Grashof, a 19th-century German mechanical engineer and a founder of the Association of German Engineers (VDI). He did not propose the number himself; later researchers named it in his honour. The Rayleigh number honours Lord Rayleigh, who in 1916 derived the condition for convection onset in a fluid layer heated from below. Rayleigh is also known for acoustics and light scattering (why the sky is blue), and won the Nobel Prize in Physics for discovering argon.
Worked examples
Example 1: differences between fluids
A 0.1 m vertical plate, 20 K temperature difference; approximate representative properties, with heat transfer coefficients from the Churchill–Chu correlation:
| Fluid | Gr | Pr | Ra | h |
|---|---|---|---|---|
| Air (35 °C) | 2.3×10⁶ | 0.71 | 1.7×10⁶ | 5 W/(m²·K) |
| Water (35 °C) | 1.2×10⁸ | 4.8 | 5.8×10⁸ | 785 W/(m²·K) |
| Engine oil (60 °C) | 2.0×10⁴ | 1,050 | 2.0×10⁷ | 73 W/(m²·K) |
Oil's Grashof number is two orders smaller than air's, yet its heat transfer coefficient is 15 times higher.
That's the Grashof–Rayleigh difference. Oil is very viscous, so it moves reluctantly and its Grashof number is small. But with a Prandtl number of about 1,000, its thermal boundary layer is much thinner than the velocity layer, so even slight flow carries heat efficiently — and oil conducts five times better than air. Heat transfer must therefore be judged by Rayleigh. Water both moves easily and conducts well, giving over 150 times air's coefficient — the reason liquid cooling is so powerful.
Example 2: judging with forced air flow
| Velocity | Re | Ri | Regime |
|---|---|---|---|
| 0.1 m/s | 606 | 6.37 | Mixed |
| 0.5 m/s | 3,030 | 0.25 | Mixed |
| 2.0 m/s | 12,121 | 0.02 | Forced dominant |
For a 0.1 m plate at 20 K, even a 0.5 m/s breeze leaves natural convection significant. Estimating temperatures where fan flow is weak, or when a fan stops, requires considering both.
Practical cautions
- Characteristic length: height for vertical plates, area ÷ perimeter for horizontal plates, diameter for cylinders, wall spacing for enclosures — follow each correlation's definition.
- Property temperature: evaluate at the film temperature; take special care with fluids like oil whose viscosity varies strongly with temperature.
- Transition: on vertical plates flow turns turbulent above Ra ≈ 10⁹; with the cube of length, large equipment reaches it quickly.
“Oil temperature doubles the result”
Computing natural convection in transformer oil, the heat transfer coefficient changed greatly depending on whether oil was taken at 40 °C or 80 °C — oil viscosity varies several-fold with temperature. Lower viscosity raises Grashof quadratically and Rayleigh substantially. Transformer oil warms in service, so cold-state properties underestimate cooling, while hot-state properties overestimate it at start-up. Include temperature-dependent properties and evaluate each operating state.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Mixing up Gr and Ra | Heat transfer badly wrong | Check the correlation's variable |
| Wrong characteristic length | Error amplified by the cube | Follow the correlation |
| Judging high-Pr fluids by Gr | Heat transfer underestimated | Use Ra |
| Treating weak flow as forced only | Wrong heat transfer | Check Ri for mixed convection |
| Constant properties (oil) | Large errors across operating states | Use temperature dependence |
I'd like to compute these dimensionless groups myself.
Try the Grashof number calculator for Grashof numbers, the natural convection simulator and Nusselt correlation tool for natural-convection heat transfer and the forced convection simulator for forced convection. Shape-specific correlations are covered in natural convection on vertical plates, natural convection in enclosures and Rayleigh–Bénard convection, and combined effects in mixed convection.
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