Natural Convection from a Sphere
Theory: buoyant flow around a sphere
Overview
When a hot sphere cools with no wind, how large is the heat transfer coefficient?
Air around the sphere warms, becomes lighter and rises — a self-generated flow, natural convection. The flow is weak, so h in air is small, a few to about 20 W/m²K, and radiation of similar size becomes relatively important. Let's compute natural-convection h with Churchill's correlation and, comparing with radiation, calculate the cooling of a hot steel ball and the heat loss from a large spherical hot-water tank.
Churchill's correlation
Properties are evaluated at the film temperature, with gas expansion coefficient β as its reciprocal.
Radiative heat transfer coefficient
With emissivity 0.8, a 300 °C surface in 20 °C surroundings gives about 16 W/m²K — comparable to natural convection.
Joining correlations
Stuart Churchill in the US systematised, in the 1970s, a way to join smoothly correlations valid at low and at high values. The '2 + …' and '[1 + (…)^n]^m' forms common in natural-convection correlations come from that idea. For spheres, it joins the Ra → 0 limit Nu = 2 with the large-Ra limit proportional to Ra^(1/4). The approach is still used in many fluid and heat correlations.
Worked examples
Example 1: Rayleigh and Nusselt numbers (Pr 0.71)
| Rayleigh number | Nusselt number |
|---|---|
| 1 | 2.45 |
| 10² | 3.44 |
| 10⁴ | 6.54 |
| 10⁶ | 16.4 |
| 10⁸ | 47.5 |
| 10¹⁰ | 146 |
Example 2: cooling a 10 mm steel ball (300 → 50 °C, 20 °C air)
| Cooling mode | h at 300 °C | Time to 50 °C |
|---|---|---|
| Natural convection only | 20.8 W/m²K | 818 s |
| Natural + radiation (ε 0.8) | 20.8 + 16.3 W/m²K | 561 s |
| Fan 5 m/s (no radiation) | 86.8 W/m²K | 158 s |
The ball's Rayleigh number at 300 °C is about 4,700. As temperature falls so does h, so the cooling was integrated in time.
Example 3: a 1 m spherical tank (60 °C, 20 °C air)
| Item | Value |
|---|---|
| Rayleigh number | 3.0×10⁹ |
| Natural-convection h | 2.95 W/m²K |
| Convective loss | 371 W |
| Radiative loss (ε 0.9) | 791 W |
| Total (per day) | 1,161 W (27.9 kWh) |
The tank loses twice as much by radiation as by natural convection.
As a sphere grows, Nusselt rises but is divided by the diameter, so h actually falls — only about 3 W/m²K at 1 m. Radiative h depends only on temperature: even at 60 °C with emissivity 0.9 it's about 6 W/m²K. So on large vessels and building exteriors radiation often exceeds natural convection, which is why covering insulation with low-emissivity foil like aluminium cuts heat loss so much. For a small steel ball natural convection is larger, yet including radiation still shortens cooling by 30%.
Calculation tips
- Evaluate properties at film temperature, updating them in time for cooling problems.
- Almost always consider radiation along with natural convection.
- Nearby walls or objects obstruct the flow and invalidate the correlation.
- With even a slight breeze, compare with forced convection as mixed convection.
- In CFD, include buoyancy and make the surrounding domain large enough.
“The insulated tank that still cooled”
At one plant, an insulated hot-water tank lost more heat than expected. The calculation had considered only natural convection from the insulation surface. But the surface was painted black with high emissivity, and radiation carried nearly twice the convective loss. Cladding it in aluminium to lower emissivity cut the loss substantially. The smaller the natural-convection coefficient, the more radiation takes the lead.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Ignoring radiation | Heat loss halved or worse | Add radiation |
| Fixed properties | Cooling time wrong | Update at film temperature |
| Radius as length scale | Ra one-eighth | Use diameter |
| Ignoring nearby obstacles | Heat transfer overestimated | Check layout |
| Ignoring slight breeze | Cooling underestimated | Check mixed convection |
I'd like to learn related topics.
Try the natural convection tool. Related pages include forced convection around a sphere, natural convection from cylinders, vertical-plate natural convection and Grashof and Rayleigh numbers.
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