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Natural Convection Coefficient Calculator

trafolo.eu / calculators / Natural convection

Most thermal calculations start by guessing a convection coefficient somewhere between 5 and 10 W/m2/K. This derives it instead, from the surface geometry, orientation and temperature rise, using the standard Rayleigh-Nusselt correlations with air properties evaluated at the film temperature.

Buoyant boundary layer rising along a heated vertical plate hot plateboundary layerbuoyant flowLTsTa ambientRa = g beta dT L^3 / (nu alpha)
Warm air rising along the surface is what carries the heat away, so the coefficient depends on the temperature rise itself - h is not a constant of the geometry.

For a vertical plate the characteristic length is the height. For horizontal surfaces it is area divided by perimeter.

Inputs — enter your values

Height for a vertical plate; area over perimeter for a horizontal one.

Surface minus ambient. The coefficient depends on it, which is why h is not a constant.

Used only to turn the coefficient into a thermal resistance and a heat flow.

Results — computed for you

Use in Temperature rise →

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Below about 1e9 the boundary layer is laminar. Above 1e9 it turns turbulent and h stops falling with size.

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Air properties are evaluated here, midway between surface and ambient.

Choosing something to solve for makes Convection coefficient an editable target and computes the chosen input from it.

Formula

Ra = g beta dT L^3 / (nu alpha), beta = 1/Tfilm Vertical (Churchill-Chu, all Ra): Nu = { 0.825 + 0.387 Ra^(1/6) / [1 + (0.492/Pr)^(9/16)]^(8/27) }^2 Horizontal, hot face up: Nu = 0.54 Ra^(1/4) below Ra 1e7 Nu = 0.15 Ra^(1/3) above Ra 1e7 Horizontal, hot face down: Nu = 0.27 Ra^(1/4) h = Nu k / L
Ra Rayleigh number, buoyancy against viscous and thermal diffusion
Nu Nusselt number, convective over conductive transfer
Pr Prandtl number of air, about 0.7
L characteristic length (m)
beta thermal expansion coefficient, 1/T for an ideal gas

What this model assumes, and where it stops

Assumptions

  • Air at one atmosphere, properties from a standard table evaluated at the film temperature.
  • An isothermal flat plate in an unbounded quiescent fluid.
  • Steady state, no radiation - radiation is handled separately on the thermal resistance page.
  • No forced airflow of any kind, including incidental draughts.
  • The surface is smooth and unobstructed on the side being considered.

Limitations

  • Real enclosures are not unbounded. Put the surface in a box and the recirculating air raises the local ambient, cutting the effective coefficient well below this figure.
  • The correlations are for isothermal plates. A real heat sink base has a temperature gradient across it, and its fins interact.
  • Nearby surfaces interfere. Two plates a few millimetres apart do not each behave like an isolated plate.
  • The horizontal correlations are power-law fits valid only over the Rayleigh bands stated; the page flags when you leave them.
  • Radiation is often comparable to natural convection at these temperatures. Ignoring it underestimates the total heat removed by a third or more.
  • The coefficient depends on the temperature rise, which usually depends on the coefficient. For a real problem you have to iterate, or solve both together.

When you need a 3D field solution instead

Closed-form models like the one above hold on idealised geometry. These are the cases where they stop being good enough and a full 3D electromagnetic and thermal solution is the only way to get a trustworthy answer:

  • Anything inside an enclosure, where the air recirculates and the far-field ambient is not what the surface sees.
  • Finned or extended surfaces, where the boundary layers of adjacent fins overlap and the isolated-plate picture fails.
  • Surfaces with a strong temperature gradient rather than a single uniform temperature.
  • Mixed convection, where a small amount of forced airflow competes with buoyancy - neither correlation set applies cleanly.
  • Cases where you need the local coefficient distribution rather than a plate average, for instance to find a hot corner.
Try Trafolo 3D FEM Talk to us about your design

Common questions

What convection coefficient does still air actually give?

For a vertical surface at 30 K above ambient: about 7.8 W/(m2.K) at 25 mm tall, 5.6 at 100 mm and 4.6 at 400 mm. The h falls roughly as the fourth root of size, so small components are cooled disproportionately well per unit area. The 5 to 10 band quoted in datasheets is real but only for hand-sized surfaces.

Why does h depend on the temperature difference?

Because the temperature difference drives the flow. Buoyancy is proportional to the density deficit of the warm air, so a hotter surface pulls a faster boundary layer past itself and h grows roughly as the fourth root of the rise. This is why natural-convection problems iterate: h depends on the rise, and the rise depends on h. The thermal resistance page does that loop for you.

Does orientation matter?

Considerably. A hot plate facing up convects freely, the same plate facing down traps its own warm air and manages roughly half the coefficient, and a vertical plate sits in between. The correlations on this page switch with the orientation selector - for fin arrays the channel spacing matters more than any of this, which is what the heat sink page is for.

References

  • Churchill & Chu - Correlating equations for laminar and turbulent free convection from a vertical plate, Int. J. Heat Mass Transfer 18 (1975)
  • Incropera & DeWitt - Fundamentals of Heat and Mass Transfer, 6th ed., ch. 9 and table A.4

Related calculators

  • Thermal resistance - Junction-to-ambient thermal resistance through a conduction path to a surface losing heat by convection and radiation in parallel.
  • Forced convection - Convection coefficient for air blown along a flat surface, from Reynolds and Nusselt, with the laminar to turbulent transition handled explicitly.
  • Heat sink fins - Fin efficiency and thermal resistance of a straight rectangular fin array, showing where extra fin height stops buying anything.
  • Busbar ampacity - Conductor temperature of a busbar in still air, solving loss, convection and radiation together instead of assuming a current density.