Convection Correlations

Convection coefficients come from empirical correlations, and each is only valid over the range it was fitted. Used outside it they do not fail loudly — they return a plausible wrong number. Even used correctly, expect roughly ±20–25% uncertainty, so treat h as a band and check the design still works at the pessimistic end.

Case	Correlation	Valid range	Notes
Flat plate, laminar	Nu = 0.664 Re^½ Pr^⅓	Re < 5×10⁵, Pr ≥ 0.6	L = distance from the leading edge; average over that length
Flat plate, turbulent	Nu = (0.037 Re^⅘ − 871) Pr^⅓	5×10⁵ < Re < 10⁸	Mixed boundary layer — laminar then turbulent
Tube, laminar, const. T_s	Nu = 3.66	Re < 2300, fully developed	Constant — h does not improve with velocity
Tube, laminar, const. q″	Nu = 4.36	Re < 2300, fully developed	Constant heat flux instead of constant wall temperature
Tube, turbulent (Dittus-Boelter)	Nu = 0.023 Re^⅘ Pr^n	Re > 10⁴, 0.6 < Pr < 160, L/D > 10	n = 0.4 heating, 0.3 cooling; L = inside diameter
Cylinder crossflow (Churchill-Bernstein)	Nu = 0.3 + 0.62Re^½Pr^⅓ / [1+(0.4/Pr)^⅔]^¼ × [1+(Re/282000)^⅝]^⅘	Re·Pr > 0.2, all Re	L = diameter; one correlation across the whole range
Sphere (Whitaker)	Nu = 2 + (0.4Re^½ + 0.06Re^⅔) Pr^0.4	3.5 < Re < 7.6×10⁴	Nu → 2 in a stagnant fluid — pure conduction
Vertical plate, natural (Churchill-Chu)	Nu = {0.825 + 0.387Ra^⅙ / [1+(0.492/Pr)^{9/16}]^{8/27}}²	all Ra	L = vertical height
Horizontal cylinder, natural	Nu = {0.60 + 0.387Ra^⅙ / [1+(0.559/Pr)^{9/16}]^{8/27}}²	Ra < 10¹²	L = diameter; the bare hot pipe case
Hot plate facing up	Nu = 0.54 Ra^¼ (Ra < 10⁷), 0.15 Ra^⅓ above	10⁴ < Ra < 10¹¹	L = area/perimeter; buoyancy assists
Hot plate facing down	Nu = 0.52 Ra^⅕	10⁴ < Ra < 10⁹	Buoyancy is trapped — much lower h

h = Nu·k/L. Evaluate fluid properties at the film temperature (surface + fluid)/2 for external flow, and at the bulk mean temperature for internal flow. Run any of these in the Convection Coefficient tool, which reports whether your case falls inside the valid range.

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