Natural Convection-Conduction Coupling
Theory: conduction, convection and radiation in series and parallel
Overview
Double glazing insulates with an air layer. Is thicker always better?
Up to a point. Still air conducts poorly, so thicker insulates better — if it doesn't move. But in a thicker gap, air rises along the warm pane and sinks along the cold one, and the circulation starts carrying heat: natural convection. Its strength depends on the cavity temperature difference, which in turn depends on how heat passes through glass and indoor/outdoor films. So conduction and convection must be solved together. Radiation across the gap is large too. Let's couple all three and find the optimum gap.
Convection in the cavity
$L$ is gap thickness, $\Delta T$ the pane-to-pane difference. The Nusselt number $Nu$ is how many times convection multiplies conductive heat transfer. In tall cavities $Nu \approx 1$ (conduction only) until the Rayleigh number exceeds a few thousand, after which convection takes effect. ElSherbiny and co-workers' vertical-cavity correlation is used here.
U-value
Indoor film, conduction through two panes, the cavity (convection and radiation in parallel) and outdoor film are in series. The cavity temperature difference depends on U, so it's found iteratively.
Rayleigh and Bénard
In 1900 France's Bénard observed neat hexagonal convection cells in a thin liquid layer heated from below. In 1916 Britain's Lord Rayleigh explained them theoretically, showing the liquid stays still until the temperature difference exceeds a threshold, after which convection begins. The dimensionless number setting that threshold is today's Rayleigh number. That convection starts in a glazing cavity beyond a certain thickness follows the same idea.
Worked example: 1 m tall double glazing
Indoor 20 °C (film 8 W/m²K), outdoor 0 °C (23 W/m²K), 4 mm glass (1.0 W/mK), emissivity 0.84 (0.10 for a low-e coating).
Gap thickness and U-value
| Gap | Plain, air | Low-e, air | Low-e, argon |
|---|---|---|---|
| 6 mm | 3.30 | 2.56 | 2.11 |
| 12 mm | 2.87 | 1.78 | 1.46 |
| 16 mm | 2.76 | 1.65 | 1.39 |
| 20 mm | 2.75 | 1.65 | 1.41 |
| 30 mm | 2.78 | 1.71 | 1.45 |
| Optimum gap | 16.3 mm (2.75) | 14.6 mm (1.65) | 15.9 mm (1.39) |
Units are W/m²K; single glazing is 5.80 W/m²K. For plain glass with air, the cavity Nusselt number stays at 1.00 up to 12 mm, then 1.30 at 20 mm and 2.08 at 30 mm as convection strengthens.
Heat split in a 12 mm gap
| Glass | Convection (conduction) | Radiation | Radiative share |
|---|---|---|---|
| Plain | 2.09 W/m²K | 3.73 W/m²K | 64% |
| Low-E | 2.10 W/m²K | 0.51 W/m²K | 19% |
Thickening stops helping around 16 mm, and over 60% of the heat was radiation anyway.
With conduction alone, heat transfer is inversely proportional to thickness, so thicker is better. But the Rayleigh number grows with the cube of thickness, so convection begins at some gap and cancels the gain — hence an optimum. Radiation barely depends on the gap and is set by emissivity; with plain glass it's the larger part, so a low-e coating does more than tuning thickness. Argon conducts less than air and lowers U further. The inner glass surface rises from 5.5 °C (single) to 12.8 °C (plain double) and 16.5 °C (low-e with argon), reducing condensation.
Analysis tips
- First compute the Rayleigh number to decide whether conduction suffices or convection must be solved.
- In CFD, include buoyancy (Boussinesq approximation) and refine near walls.
- Always include radiation in cavities and enclosures; it can equal or exceed convection.
- Small temperature differences give weak flow and slow convergence; watch relaxation and initial values.
- At large Rayleigh numbers (about 10⁹ and above) flow becomes turbulent; consider a turbulence model.
“A thicker gap, yet still cold”
In a house renovation, double glazing with a custom 30 mm gap replaced a 12 mm one to improve insulation, yet comfort barely changed. The calculation shows U for plain glass only falls from 2.87 to 2.78 W/m²K — 3% — because convection had started in the gap and much of the heat was radiation anyway. Switching to low-e glass greatly reduced the chill near the windows. Without considering natural convection and radiation together, the effective fix isn't chosen.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Treating the gap as conduction only | Thick gaps overrated | Check Rayleigh number |
| Ignoring radiation | Heat loss badly underestimated | Include radiation |
| Forgetting buoyancy | No flow develops | Set gravity and expansion |
| Fixing the cavity ΔT | Coupling ignored | Iterate |
| Coarse near-wall mesh | Heat-transfer error | Resolve boundary layer |
I'd like to learn related topics.
Try the natural convection tool and convection–diffusion simulator. Related pages include conjugate heat transfer, mixed convection, thermal plumes and thermal stratification.
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