Rayleigh Number Calculator

Ra = gβΔTL³/(να) = Gr × Pr — governs natural convection onset and intensity. Used directly in Nu correlations (Churchill-Chu, Morgan, etc.) for heat transfer coefficient prediction.

Heat Transfer · Natural Convection

Inputs

Solve for:

Fluid preset (fills β, ν, α, k):

Thermal expansion coefficient β (1/K)

Ideal gas: β = 1/T_film (K). Liquids: use tabulated values.

Temperature difference ΔT (K)

|T_wall − T_∞| (magnitude)

Characteristic length L (m)

Kinematic viscosity ν (m²/s)

Thermal diffusivity α (m²/s)

α = k / (ρ c_p) — Pr = ν / α is computed automatically

Also compute heat transfer coefficient h — enter thermal conductivity k

Theory

Definition — three equivalent forms:

Ra = g × β × ΔT × L³ / (ν × α) [from diffusivities]

Ra = Gr × Pr [Grashof × Prandtl]

α = k / (ρ × c_p) [thermal diffusivity]

Pr = ν / α [derived automatically]

Churchill-Chu correlation (vertical isothermal plate, all Ra):

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

h = Nu × k / L

Valid for all Ra from laminar to turbulent. For laminar only (Ra < 10⁹) a slightly more accurate form exists: Nu = 0.68 + 0.670 Ra^(1/4) / [1 + (0.492/Pr)^(9/16)]^(4/9).

Regime thresholds (vertical plate):

Ra < 10⁴ negligible natural convection — conduction dominates

10⁴ – 10⁹ laminar natural convection boundary layer

Ra > 10⁹ turbulent natural convection

Characteristic length L by geometry:

Geometry	L	Correlation
Vertical plate / wall	plate height H	Churchill-Chu
Horizontal cylinder	diameter D	Morgan; Churchill-Bernstein
Sphere	diameter D	Churchill
Horizontal plate (hot up / cold down)	A/P	McAdams
Horizontal plate (hot down)	A/P	McAdams (weaker convection)

Ra vs Gr vs Pr:

Gr = g β ΔT L³ / ν² (buoyancy vs viscous — no thermal info)

Pr = ν / α (momentum vs thermal diffusivity)

Ra = Gr × Pr (full natural-convection parameter)

Nu correlations almost always require Ra, not Gr alone. Always multiply by Pr before applying a heat transfer correlation.