Engineering & Science
Churchill-Chu Vertical Plate Natural Convection Formula (All Rayleigh Numbers)
Calculate free-convection heat transfer Nu_L = [0.825 + 0.387*Ra^(1/6) / (1 + (0.492/Pr)^(9/16))^(8/27)]^2 across laminar and turbulent buoyancy boundary layers.
One unified correlation spanning 13 orders of magnitude of Rayleigh number (10^-1 < Ra_L < 10^12)!
Before Stuart W. Churchill and Humbert H. S. Chu (1975, International Journal of Heat and Mass Transfer), engineers had to switch discontinuously at Ra_L = 10^9 between Ostrach’s laminar Ra^(1/4) boundary-layer solution and turbulent Ra^(1/3) plume correlations! Applying Churchill-Usagi asymptotic power blending, Churchill & Chu unified the ENTIRE laminar-to-turbulent isothermal vertical-wall regime into Nu_L = [0.825 + 0.387 * Ra_L^(1/6) / (1 + (0.492 / Pr)^(9/16))^(8/27)]^2!
Notice how squaring Ra_L^(1/6) naturally recovers the turbulent Ra_L^(1/3) law at high Rayleigh numbers (where h becomes independent of wall height L!) while smoothly blending into 0.68 + laminar conduction at low Ra_L! For a heated electronics cabinet wall in air (Pr = 0.71) at Ra_L = 2.5×10^8, Nu_L = 80.42—jumping to Nu_L = 252.28 for a tall industrial vessel at turbulent Ra_L = 1.0×10^10!
| Vertical Wall Rayleigh Number Ra_L (Air Pr = 0.71) | Boundary-Layer Regime | Churchill-Chu Nu_L | Engineering Application |
|---|---|---|---|
| Ra_L = 2.5×10^8 (0.6 m Cabinet Panel) | Upper Laminar (< 10^9) | 80.42 | Passive heatsink / enclosure cooling |
| Ra_L = 1.0×10^10 (2.2 m Tank Wall) | Turbulent Plume (> 10^9) | 252.28 | Storage tank ambient heat loss |
| Invalid Ra_L <= 0 or Pr <= 0 | Error Guard | "Invalid Rayleigh or Prandtl" | Thermophysical Bounds Guard |
Churchill-Chu All-Rayleigh Vertical Plate Nusselt Number Nu_L
Evaluates [0.825 + 0.387 * Ra^(1/6) / (1 + (0.492/Pr)^(9/16))^(8/27)]^2 from Rayleigh number B2 and Prandtl number B3.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Rayleigh or Prandtl", ROUND((0.825 + (0.387 * (B2 ^ (1 / 6))) / ((1 + ((0.492 / B3) ^ (9 / 16))) ^ (8 / 27))) ^ 2, 2))Prandtl Universal Damping Function ψ(Pr) = [1 + (0.492 / Pr)^(9/16)]^(-8/27)
Isolates the Prandtl-number modifier bridging liquid metals (Pr << 1), gases (Pr ≈ 0.71), and viscous oils (Pr >> 1).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Rayleigh or Prandtl", ROUND((1 + ((0.492 / B3) ^ (9 / 16))) ^ (-8 / 27), 4))Try the example
Use the original values or change them to test your understanding.