Natural Convection on Vertical Plate
Theory: warm air rising along the plate
Overview
I'm designing a fanless power-supply enclosure and want to estimate how much heat leaves through the side panels.
Air touching the warm plate becomes lighter and rises along it, drawing cooler air in from below to be heated in turn. That is natural convection; the thermal boundary layer thickens from the bottom edge upward. The only driving force is buoyancy, so the flow is slow and transports heat far less effectively than forced convection — which is why you need the numbers.
Rayleigh number
$g$ is gravity, $\beta$ the expansion coefficient (the reciprocal of absolute temperature for air), $\Delta T$ the plate-to-ambient difference, $H$ the plate height, $\nu$ kinematic viscosity and $\alpha$ thermal diffusivity. It compares buoyancy with viscous and thermal diffusion and scales with the cube of height. Above about 10⁹ the flow becomes turbulent.
Churchill–Chu correlation
It gives the plate-averaged Nusselt number from laminar to turbulent in one expression, for an isothermal plate with properties at the film temperature (mean of plate and ambient). For laminar flow the simpler $\overline{Nu} = 0.59\,Ra^{1/4}$ is also common.
Research born from room radiators
Natural convection on vertical plates has been studied since the early 20th century, motivated by room radiators and the cooling of electrical equipment. In the 1930s, detailed measurements of air temperature and velocity near a plate were explained with boundary-layer theory. In 1975 Churchill and Chu condensed many experiments into a single curve; usable from laminar to turbulent, it remains the textbook and design standard. Room “radiators”, incidentally, move most of their heat by natural convection rather than radiation.
Worked examples
Example 1: coefficient by height and temperature difference
Isothermal vertical plate in 25 °C air, from Churchill–Chu with properties at the film temperature:
| ΔT | Height | Ra | h | Heat flux |
|---|---|---|---|---|
| 10 K | 0.05 m | 1.1×10⁵ | 5.0 W/(m²·K) | 50 W/m² |
| 10 K | 0.2 m | 7.0×10⁶ | 3.7 W/(m²·K) | 37 W/m² |
| 30 K | 0.05 m | 2.8×10⁵ | 6.5 W/(m²·K) | 194 W/m² |
| 30 K | 0.2 m | 1.8×10⁷ | 5.0 W/(m²·K) | 151 W/m² |
| 30 K | 1.0 m | 2.2×10⁹ | 4.3 W/(m²·K) | 129 W/m² |
| 60 K | 0.2 m | 2.9×10⁷ | 6.0 W/(m²·K) | 360 W/m² |
The shorter plate has the higher coefficient.
In laminar flow the boundary layer thickens with height, so heat passes less easily. The average coefficient goes as height to the −1/4 power, so for the same area, short plates side by side beat one tall plate slightly. Still, everything here is 3–8 W/(m²·K), an order of magnitude below the tens of W/(m²·K) with a fan. A 0.2 m square plate at 30 K sheds only about 6 W per side.
Example 2: transition height and boundary-layer thickness
| ΔT | Height at Ra = 10⁹ |
|---|---|
| 10 K | 1.05 m |
| 30 K | 0.76 m |
| 60 K | 0.65 m |
At 30 K the laminar boundary layer is about 11 mm thick 0.05 m from the bottom edge and about 15 mm at 0.2 m. If adjacent plates are closer than about twice this, the boundary layers merge, the flow chokes and heat transfer drops — a guide for heat-sink fin spacing.
Example 3: comparison with radiation
| ΔT | Radiation (ε = 0.9) | Natural convection (0.2 m) |
|---|---|---|
| 10 K | 5.7 W/(m²·K) | 3.7 W/(m²·K) |
| 30 K | 6.3 W/(m²·K) | 5.0 W/(m²·K) |
| 60 K | 7.3 W/(m²·K) | 6.0 W/(m²·K) |
On a painted surface (emissivity about 0.9), radiation carries as much heat as natural convection or more. On polished aluminium (about 0.05) it contributes almost nothing — which is why fanless equipment is often painted black.
How to approach the analysis
CFD cautions
- Use a large domain: leave ample space above and to the sides so warm plume air does not recirculate, and use open (ambient-pressure) boundaries.
- Resolve the boundary layer: place at least 10 cells across a boundary layer about 10 mm thick.
- Buoyancy model: the Boussinesq approximation is fine for small differences; beyond several tens of K, make density a function of temperature.
- Include radiation: never omit it with natural convection in air.
Checking results
- Estimate h with Churchill–Chu first and compare with the CFD average.
- Confirm results don't change when the domain is enlarged.
- Check the energy balance: heat input = convection + radiation.
“It ran hotter once boxed”
A board that was fine on the bench runs over 10 K hotter inside a box with small vents — a common natural-convection story. The flow around a plate depends on cool air entering below and warm air leaving above; small inlets and outlets choke it, and warm air pools inside. Put intakes low and exhausts high, with a large height difference, and the chimney effect strengthens the flow. The analysis must include air outside the box to capture this.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Omitting radiation | Heat loss underestimated by nearly half | Include emissivity |
| Using forced-convection coefficients | Temperatures badly underestimated | Confirm 3–8 W/(m²·K) with natural-convection correlations |
| Domain too small | Warm air recirculates, temperature rises | Enlarge upward, use open boundaries |
| Fins too close | Boundary layers merge and choke | Keep about twice the boundary-layer thickness |
| Properties at ambient temperature | Errors of a few percent | Use the film temperature |
I'd like to try the numbers myself.
Try the natural convection simulator and Nusselt correlation tool for coefficients, the Grashof number calculator for the dimensionless groups, and the finned heat sink calculator for fins. The meaning of the groups is explained in Grashof and Rayleigh numbers, and other orientations and shapes in natural convection on horizontal plates and natural convection in vertical channels.
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