Forced Convection around a Sphere
Theory: correlations for spheres
Overview
There seem to be several sphere heat-transfer correlations. Which should I use?
The standard ones are Whitaker's and Ranz–Marshall's. Both start from Nu = 2 (conduction into the surroundings with no flow) and add a flow term. Up to about Re = 1000 they nearly coincide, but they diverge at higher Reynolds numbers. Ranz–Marshall came from droplet-evaporation experiments and suits small particles and droplets, while Whitaker fits larger spheres in faster flow. Let's check with a cooling steel ball and fine particles.
The two correlations
Whitaker's correlation covers Re = 3.5 to 7.6×10⁴ and Pr = 0.71 to 380. μ/μ_s is the ratio to viscosity at wall temperature, close to 1 for gases.
Cooling time by lumped capacitance
A correlation born from spray drying
The Ranz–Marshall correlation was built in 1952 by Ranz and Marshall in the US, who measured evaporation of suspended water drops for spray-dryer design. Droplet evaporation and heat transfer share the same form, so it's still used wherever particles or droplets appear: fuel sprays, spray drying, powder heating. Whitaker's correlation of 1972 re-fitted a wide body of earlier data to cover broad Reynolds and Prandtl ranges.
Worked examples
Example 1: comparing the correlations (air, Pr 0.71)
| Reynolds number | Whitaker | Ranz-Marshall | Difference |
|---|---|---|---|
| 1 | 2.40 | 2.54 | +6% |
| 10 | 3.35 | 3.69 | +10% |
| 100 | 6.62 | 7.35 | +11% |
| 1,000 | 18.3 | 18.9 | +4% |
| 10,000 | 61.2 | 55.5 | −9% |
| 70,000 | 183 | 144 | −22% |
Example 2: air-cooling a 10 mm steel ball
A 300 °C steel ball cooled to 50 °C in 20 °C air (Whitaker, no radiation). At 5 m/s: Re = 3185, Nu = 33.0, h = 86.8 W/m²K, Bi = 0.0032, time constant 71 s.
| Air speed | h | Time to 50 °C |
|---|---|---|
| 0.5 m/s | 28.0 W/m²K | 490 s |
| 2 m/s | 54.2 W/m²K | 254 s |
| 5 m/s | 86.8 W/m²K | 158 s |
| 10 m/s | 126 W/m²K | 109 s |
| 20 m/s | 184 W/m²K | 75 s |
| 40 m/s | 271 W/m²K | 51 s |
Even at eight times the speed, 40 m/s, cooling only takes a third as long.
The boundary layer thins with speed, but h grows only about as the square root of speed. So halving cooling time needs 3.6 times the speed — from 5 to about 18 m/s — and since fan power rises with the cube of speed, efficiency worsens fast. At low speed, radiation (not included here) matters too: at emissivity 0.8 it's about 16 W/m²K at 300 °C, not negligible against 28 W/m²K at 0.5 m/s. With Bi = 0.003, internal temperature differences in the ball can be ignored.
Example 3: fine particles in still air (Nu = 2)
| Particle diameter | h = 2k/d |
|---|---|
| 50 µm | 1,050 W/m²K |
| 5 µm | 10,500 W/m²K |
Calculation tips
- Evaluate properties at film temperature and apply the viscosity correction.
- Check the Reynolds number is within the correlation's range.
- At low speeds add natural convection and radiation.
- If Bi exceeds 0.1, solve internal conduction instead of lumped capacitance.
- In particle clouds and packed beds, neighbours change heat transfer.
“Quenching bearing balls”
Bearing balls are cooled in oil or air after hardening. Smaller balls have higher heat transfer coefficients and more surface per volume, so they cool faster. Doubling diameter cuts h to about 0.7 times while the time constant grows with diameter, so cooling takes about three times longer. That's why, cooled under identical conditions, only the larger balls may lack hardness — and cooling conditions are adjusted to ball size.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Ranz–Marshall at high Re | About 20% low | Use Whitaker |
| Forgetting Nu = 2 | Too low at low Re | Include conduction term |
| Ignoring radiation | Hot cooling underestimated | Add radiation |
| Unchecked Bi | Internal gradients missed | Check Bi < 0.1 |
| Radius as length scale | Re halved | Use diameter |
I'd like to learn related topics.
Try the forced convection tool. Related pages include cylinder in crossflow, flat plate external flow, lumped capacitance and natural convection from spheres.
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