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Vertical Plate — Isothermal

Average h and heat rate from an isothermal vertical plate in natural convection.

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Orientation alone can halve your cooling: the same plate at the same ΔT convects about twice as well with its hot face up as with its hot face down.

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Vertical Plate in Natural Convection — Isothermal

Calculates the average convective heat transfer coefficient and total heat rate for a vertical plate held at a uniform surface temperature in an otherwise quiescent fluid. Buoyancy drives a free-convection boundary layer up the heated face. The plate is assumed to exchange heat from one side only.

Rayleigh Number

Fluid properties are evaluated at the film temperature Tf=(Ts+T)/2. The Rayleigh number uses the plate height as the characteristic length:

RaH=ναgβ(TsT)H3=ν2gβ(TsT)H3Pr

where β is the volumetric thermal expansion coefficient, ν the kinematic viscosity and α=ν/Pr the thermal diffusivity.

Nusselt Number (Churchill–Chu)

For the laminar regime (RaH<109):

Nu=0.68+[1+(0.492/Pr)9/16]4/90.670RaH1/4

For the turbulent regime (RaH109) the all-regime form is used:

Nu={0.825+[1+(0.492/Pr)9/16]8/270.387RaH1/6}2

Heat Transfer Coefficient and Heat Rate

h=HNuk,Q=hA(TsT),A=HW

Validity

  • Valid across the full laminar-to-turbulent range of Rayleigh numbers.
  • Applies to gases and liquids with Prandtl numbers from roughly 0.01 upward.

References

  • Churchill, S. W. and Chu, H. H. S., Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate, Int. J. Heat Mass Transfer, 18, 1323, 1975.
  • Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, eqs. 9.26 & 9.27.
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Frequently asked questions

Average h and heat rate from an isothermal vertical plate in natural convection. Enter your inputs and press Calculate — the worked solution shows every step of the method with your numbers substituted in.

When there is no fan, pump or significant draught — enclosures, outdoor surfaces on still days, passive heat sinks. If there is any imposed flow, compare the two mechanisms: when Gr/Re² is well below one, forced convection dominates and the forced-convection calculators are the right tools.

No — it computes the convective part only. At typical natural-convection coefficients of 2–10 W/m²·K, radiation from a painted or oxidised surface is often the same order of magnitude, so total heat loss should add a radiative term from the Radiation calculators.