Natural Convection around a Cylinder
Theory: the plume around a cylinder
Overview
Some insulation has come off a hot-water pipe at the plant. I'd like to know how much heat escapes per metre.
Around a horizontal pipe, warmed air flows from underneath up both sides and leaves as a plume above. That's natural convection from a cylinder, and the coefficient depends on the Rayleigh number based on diameter. At pipe temperatures radiation contributes about as much, so the key is to add both.
The Churchill–Chu correlation
Properties are evaluated at the film temperature (the mean of surface and ambient), and for an ideal gas $\beta = 1/T_f$. A single expression covers laminar through turbulent conditions, which is why it is the most widely used correlation for horizontal cylinders.
Adding radiation
One formula for everything
Natural-convection correlations used to be split into separate laminar and turbulent formulas. In 1975 Churchill and Chu systematically blended expressions that match each limit — very low and very high Rayleigh number — with a power-law sum, producing single formulas for vertical plates and horizontal cylinders. With no jump at the transition, they are easy to use in design calculations and code. Many correlations in use today follow the same idea of joining asymptotes.
Worked example: hot-water pipe heat loss and insulation
Bare pipe
A 60.3 mm outside-diameter pipe at 80 °C in 20 °C air. With air properties at the 50 °C film temperature, $Ra_D \approx 8.5\times10^{5}$, $\overline{Nu}_D \approx 13.9$ and the convective coefficient is about 6.4 W/(m²·K).
| Surface | Emissivity | Radiative coefficient | Heat loss per metre | Convection share |
|---|---|---|---|---|
| Painted | 0.9 | 7.0 W/(m²·K) | 152 W/m | 48 % |
| Shiny metal | 0.1 | 0.8 W/(m²·K) | 82 W/m | 89 % |
A painted bare pipe loses about 150 W per metre — about 1,300 kWh per metre over a year of continuous operation. Ten metres of stripped insulation throws away as much heat as several households use in electricity.
With insulation
Wrapping 25 mm of glass wool (k = 0.04 W/(m·K)) and letting the jacket (emissivity 0.3) lose heat by natural convection and radiation, iterating for the jacket temperature gives a loss of about 20.5 W/m with the jacket at about 31 °C — roughly 87% less than the painted bare pipe.
Critical radius: when cladding increases heat loss
Insulating a cylinder adds conduction resistance but also enlarges the outer surface, reducing the surface resistance. While the outer radius is below $r_c = k/h$ the second effect wins and loss grows with thickness. For glass wool on a pipe $r_c \approx 4$ mm and it doesn't matter, but for a wire clad in PVC ($k = 0.16$) $r_c \approx 13$ mm. In a simplified calculation with $h$ fixed at 12 W/(m²·K), a 2 mm conductor 40 K above ambient dissipates:
| Cladding thickness | Heat dissipation per metre |
|---|---|
| None | 3.0 W/m |
| 0.5 mm | 4.3 W/m |
| 2 mm | 7.3 W/m |
| 5 mm | 10.0 W/m |
| 12 mm | 11.2 W/m |
| 30 mm | 10.4 W/m |
Cladding cools it more than three times better! I thought wire insulation was only for electrical isolation.
It does for thin wires. In reality the heat-transfer coefficient also changes with diameter and the cladding has a temperature limit, so the table shows the trend only. Current ratings for cables come from standards and detailed calculations that include such effects.
Tips for CFD
Setup
- Domain size: the plume extends far above the cylinder, so allow at least ten diameters above and several to the sides and below. A tight domain pushes the plume back at the boundary and changes the coefficient.
- Boundary-layer mesh: the thermal boundary layer is thinnest on the underside. Make the first cell off the wall thin enough to place several layers inside it.
- Laminar or turbulent: below roughly $Ra_D = 10^9$ solve as laminar. Switching on a turbulence model for weak natural convection can add excessive eddy viscosity and skew the coefficient.
- Radiation: at pipe temperatures radiation can be about half of the loss, so always include a radiation model with the surface emissivity.
Verification
First compare the mean Nusselt number from a 2D model without radiation against Churchill–Chu for several temperature differences; agreement within about 10% indicates a sound setup. Then add radiation and compare with the heat losses above. When several pipes run close together, the plume from a lower pipe envelops the one above, and single-cylinder correlations no longer apply.
A trap when assessing pipe insulation
When analysing insulation, its conductivity is often entered as a constant catalogue value. In reality the conductivity of glass or rock wool rises with temperature and rises further when it absorbs moisture. On outdoor pipes, rain entering through jacket seams can degrade performance several-fold. If the analysis predicts much less loss than was measured, suspect the condition of the insulation on site before the model.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Ignoring radiation | Loss underestimated by about half for painted surfaces | Include emissivity and add radiation |
| Using pipe length as the characteristic length | Rayleigh number off by orders of magnitude | Use the diameter for horizontal cylinders |
| Applying horizontal-cylinder correlations to vertical pipes | Different flow pattern, errors | Use vertical-plate correlations (for large diameters) |
| Assuming cladding insulates a thin wire | Below the critical radius, loss actually increases | Compare the outer radius with $r_c = k/h$ |
| Using single-cylinder correlations for pipe banks | Upper pipes' loss overestimated | Use CFD or tube-bank correlations |
I'd like to try other conditions myself.
Try the natural convection coefficient calculator for heat-transfer coefficients and the critical insulation radius simulator to see how the critical radius works. For flow around a cylinder in wind, see the flow around a cylinder simulator. For other shapes see natural convection on a vertical plate or natural convection from a sphere, and for horizontal layers heated from below, Rayleigh–Bénard convection.
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