How to use this calculator
Pick the surface, enter its size and the surface and air temperatures. Air properties are evaluated automatically at the film temperature; for another fluid, enter its properties.
Formulas
Rayleigh numberRa = gβ(Ts − T∞)L³/(να), β = 1/Tf for a gas
Vertical plateNu = {0.825 + 0.387Ra1/6/[1 + (0.492/Pr)9/16]8/27}²
Horizontal cylinderNu = {0.60 + 0.387Ra1/6/[1 + (0.559/Pr)9/16]8/27}²
Heat transferh = Nu k/L, Q = hA(Ts − T∞)
Worked example
0.5 m × 0.5 m vertical plate at 70 °C in 20 °C air (the default)
- Tf = 45 °C: k = 0.0277 W/m·K, ν = 1.75 × 10⁻⁵ m²/s, Pr = 0.705, β = 1/318 K⁻¹.
- Ra = 9.81 × (1/318) × 50 × 0.5³/(να) = 4.44 × 10⁸, Nu = 95.7.
- h = 5.30 W/m²·K and Q = 5.30 × 0.25 × 50 = 66.3 W.
Frequently asked questions
Why the film temperature?
Properties vary across the boundary layer; evaluating them at the average of surface and fluid temperatures is the standard approximation in correlations.
What about a horizontal plate?
Its length scale is A/P. A hot surface facing up drives a stronger plume than one facing down, so it uses a different correlation.