Natural Convection — Vertical Plate

Churchill-Chu correlation for free convection on an isothermal vertical surface. Computes Gr, Ra, Pr, Nu, and optionally h = Nu × k / L across laminar and turbulent regimes.

Heat Transfer · Natural Convection

Inputs

Fluid preset at film temperature (fills β, ν, α, k):

Temperature units:

Wall temperature Tw (°C)

Ambient temperature T∞ (°C)

Plate height L (m)

Thermal expansion coefficient β (1/K)

Auto-compute β = 1/T_film (ideal gas)

Kinematic viscosity ν (m²/s)

Thermal diffusivity α (m²/s)

Pr = ν/α is computed automatically

Also compute h = Nu × k / L — enter fluid thermal conductivity k

Theory

Dimensionless groups:

Gr = g × β × ΔT × L³ / ν² [Grashof — buoyancy vs viscous]

Ra = Gr × Pr = g × β × ΔT × L³ / (ν × α) [Rayleigh]

Pr = ν / α [Prandtl — from inputs]

Churchill-Chu correlation:

Laminar (Ra < 10⁹):

Nu = 0.68 + 0.670 Ra^(1/4) / [1 + (0.492/Pr)^(9/16)]^(4/9)

All-range (Ra ≥ 10⁹, or preferred form for all Ra):

Nu = {0.825 + 0.387 Ra^(1/6) / [1 + (0.492/Pr)^(9/16)]^(8/27)}²

Thermal expansion coefficient β:

Ideal gas: β = 1 / T_film (T_film in Kelvin)

T_film = (T_wall + T_ambient) / 2

Liquids: use tabulated β — do NOT use ideal-gas formula

Regime thresholds:

Ra < 10⁴ negligible natural convection

10⁴ – 10⁹ laminar boundary layer on plate

Ra > 10⁹ turbulent natural convection

Orientation matters — characteristic length L:

Geometry	L	Correlation
Vertical plate / wall	height H	Churchill-Chu (this calculator)
Vertical cylinder	height H	Churchill-Chu (if D/H ≥ 35/Gr^0.25)
Horizontal cylinder	diameter D	Churchill-Bernstein
Sphere	diameter D	Churchill
Horizontal plate (hot up)	area/perimeter A/P	McAdams: Nu = 0.54 Ra^(1/4) laminar