Natural Convection in Vertical Channels
Theory: narrow channels choke the flow
Overview
A fanless heat sink cools better with more fins, right?
Up to a point — then it backfires. In natural convection, air between fins warms and rises on its own. If the gap is narrow, boundary layers from both walls meet and choke the flow, so h drops sharply; if wide, there are too few fins and too little area. In between lies an optimum spacing of maximum dissipation. Let's use the correlation for natural convection between vertical plates to find it for a 100 mm tall heat sink.
The Bar-Cohen–Rohsenow correlation
For symmetric isothermal plates. Small El (narrow) gives $Nu_S \approx El/24$, fully developed channel flow; large El (wide) gives $Nu_S \approx 0.59\,El^{1/4}$, like an isolated plate.
Optimum fin spacing estimate
Elenbaas's experiments
In 1942 Elenbaas at Philips in the Netherlands, thinking about cooling vacuum tubes and lamps, measured natural convection between facing vertical plates and organised the results with a dimensionless number including spacing and height — still called the Elenbaas number. In 1984 Bar-Cohen and Rohsenow in the US combined the narrow and wide limits into one correlation, which became widely used for naturally cooled electronics heat sinks.
Worked example: a natural-convection heat sink
Fins 100 mm tall, 50 mm deep, 1 mm thick on a 100 mm wide base; fin-to-air difference 40 K (air, film 40 °C); fins treated as isothermal. Ra_L = 3.07×10⁶.
| Spacing | Fins | Elenbaas number | h | Dissipation |
|---|---|---|---|---|
| 2 mm | 34 | 0.5 | 0.28 W/m²K | 3.7 W |
| 4 mm | 20 | 7.9 | 2.11 W/m²K | 16.0 W |
| 6 mm | 15 | 40 | 4.99 W/m²K | 28.0 W |
| 8 mm | 12 | 126 | 6.26 W/m²K | 27.6 W |
| 10 mm | 10 | 307 | 6.57 W/m²K | 23.7 W |
| 15 mm | 7 | 1,556 | 6.68 W/m²K | 16.0 W |
| 20 mm | 5 | 4,916 | 6.69 W/m²K | 10.7 W |
Searching in 0.05 mm steps gives an optimum of 6.60 mm (28.9 W), within 2% of the 6.48 mm estimate. A flat base without fins manages only about 1.3 W.
The 2 mm spacing has the most fins, yet dissipates only an eighth of the optimum.
At 2 mm the Elenbaas number is only 0.5 and air in the channels barely moves: h is 0.28 W/m²K, about a twentieth of the optimum. More than twice the area can't make up for that. Widening the gap, h tops out at the single-plate value (about 6.7 W/m²K), and beyond that dissipation falls as fins are removed. Forced-air heat sinks often have fins 1–2 mm apart, so if the fan stops, natural convection barely cools them. This calculation shows whether a design stays safe on fan failure.
Design tips
- Start with S_opt ≈ 2.714·L·Ra_L^(−1/4) as a guide.
- Orient fins vertically and keep the bottom inlet and top outlet clear.
- Account for fin efficiency; thin, long fins lose efficiency.
- Add radiation; it can be 20–30% of dissipation in natural cooling.
- In an enclosure, check vents above and below don't restrict the flow.
“It overheated when the fan stopped”
In outdoor telecom equipment, when a fan failed in summer the unit quickly overheated and shut down. The design assumed natural convection would cope for a while after fan failure, but the heat sink's fins were 2 mm apart for forced air. Under natural convection the air between fins barely moved and dissipation was a fraction of expectations. The revised design widened spacing to about 7 mm, workable in natural convection, and made fins taller to recover area.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Isolated-plate formula in channels | Overestimate at narrow gaps | Use channel correlation |
| Forced-air spacing reused | Fails in natural cooling | Design to S_opt |
| Ignoring radiation | Dissipation underestimated | Add radiation |
| Fin efficiency = 1 | Overestimate | Compute efficiency |
| Ignoring vent resistance | Flow overestimated | Model the enclosure |
I'd like to learn related topics.
Try the heat sink simulator and natural convection tool. Related pages include vertical-plate natural convection, natural convection in enclosures and heat sink design.
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