convection

name	convection
description	Convection heat transfer — forced internal/external, natural convection, boundary layers, Nusselt number correlations (Dittus-Boelter, Churchill-Bernstein, etc.), boiling, condensation.
metadata	{"priority":7,"promptSignals":{"phrases":["convection","Nusselt","heat transfer coefficient","forced convection","natural convection","boundary layer","boiling","condensation"],"minScore":4}}

Convection Heat Transfer — Complete Skill

Key Dimensionless Numbers

Re = ρVL/μ = VL/ν (Reynolds — inertia/viscous) Nu = hL/k (Nusselt — convection/conduction) Pr = μc_p/k = ν/α (Prandtl — momentum/thermal diffusivity) Gr = gβΔTL³/ν² (Grashof — buoyancy/viscous for natural convection) Ra = Gr·Pr (Rayleigh) St = Nu/(Re·Pr) = h/(ρVc_p) (Stanton)

Property evaluation: use film temperature T_f = (T_s+T_∞)/2 unless noted Viscosity correction: (μ/μ_s)^0.14 for liquids with large T variation

External Forced Convection

Flat plate (laminar, Re_L < 5×10⁵): Local: Nu_x = 0.332·Re_x^0.5·Pr^(1/3) (Pr ≥ 0.6) Average: Nu_L = 0.664·Re_L^0.5·Pr^(1/3) δ_thermal = δ_velocity·Pr^(-1/3)

Flat plate (turbulent, Re > 5×10⁵): Local: Nu_x = 0.0296·Re_x^0.8·Pr^(1/3) Mixed (transition at Re_c=5×10⁵): Nu_L = (0.037·Re_L^0.8 - 871)·Pr^(1/3)

Cylinder in crossflow (Churchill-Bernstein): Nu = 0.3 + 0.62·Re^0.5·Pr^(1/3)/(1+(0.4/Pr)^(2/3))^0.25 · (1+(Re/282000)^(5/8))^(4/5) Valid: Re·Pr > 0.2

Sphere (Whitaker): Nu = 2 + (0.4·Re^0.5 + 0.06·Re^(2/3))·Pr^0.4·(μ/μ_s)^0.25 Valid: 3.5 ≤ Re ≤ 7.6×10⁴, 0.71 ≤ Pr ≤ 380

Internal Forced Convection (Pipe/Duct)

Entry length: Hydrodynamic: L_h = 0.05·Re·D (laminar), L_h/D ≈ 10-60 (turbulent) Thermal: L_t = 0.05·Re·Pr·D (laminar)

Laminar, fully developed: Constant T_s: Nu = 3.66 Constant q''_s: Nu = 4.36 Combined entry: Nu = 3.66 + 0.0668(D/L·Re·Pr)/(1+0.04(D/L·Re·Pr)^(2/3)) (Hausen)

Turbulent, fully developed (Re > 10,000, 0.6 < Pr < 160): Nu = 0.023·Re^0.8·Pr^n (Dittus-Boelter): n=0.4 (heating), n=0.3 (cooling) Nu = (f/8)(Re-1000)Pr / (1+12.7√(f/8)(Pr^(2/3)-1)) (Gnielinski, more accurate) f = (0.790·ln(Re)-1.64)^(-2) (Petukhov)

Non-circular ducts: Use D_h = 4A_c/P (hydraulic diameter) Annulus: D_h = D_o - D_i

Temperature rise for constant heat flux: T_s(x) - T_m(x) = q''_s/h_x [constant for fully developed] T_m(x) = T_m,in + q''_s·P·x/(ṁ·c_p)

Natural Convection

Vertical plate (Churchill-Chu): Nu = (0.825 + 0.387·Ra^(1/6)/(1+(0.492/Pr)^(9/16))^(8/27))² [all Ra] β = 1/T_f [ideal gas], β ≈ 0.00341 K⁻¹ (air at 20°C)

Horizontal cylinder: Nu = (0.60 + 0.387·Ra^(1/6)/(1+(0.559/Pr)^(9/16))^(8/27))²

Sphere: Nu = 2 + 0.589·Ra^(1/4)/(1+(0.469/Pr)^(9/16))^(4/9)

Horizontal plate: Hot side up or cold side down (Ra: 10⁴-10⁷): Nu = 0.54·Ra^(1/4) Hot side up or cold side down (Ra: 10⁷-10¹¹): Nu = 0.15·Ra^(1/3) Hot side down or cold side up: Nu = 0.27·Ra^(1/4)

Mixed convection: Gr/Re² >> 1 → natural; << 1 → forced; ~1 → mixed Nu_combined = (Nu_forced^n ± Nu_natural^n)^(1/n), n=3 (vertical), ± (assist vs. oppose)

Pool Boiling (Nukiyama curve)

Free convection: q'' = h·(T_s-T_sat), h from natural convection Nucleate boiling (Rohsenow): q'' = μ_l·h_fg·(g(ρ_l-ρ_v)/σ)^0.5 · (c_pl(T_s-T_sat)/(C_sf·h_fg·Pr_l^n))³ C_sf = surface-fluid constant (0.006 for water-Cu polished) Critical heat flux (Zuber): q''_max = 0.131·ρ_v·h_fg·(σ·g(ρ_l-ρ_v)/ρ_v²)^0.25 q''_max ≈ 1.2 MW/m² for water at 1 atm

Minimum heat flux (Berenson): q''_min = 0.09·ρ_v·h_fg·(σ·g(ρ_l-ρ_v)/(ρ_l+ρ_v)²)^0.25

Film boiling: Nu_D = 0.62·(ρ_v(ρ_l-ρ_v)g·h''_fg·D³/(μ_v·k_v·(T_s-T_sat)))^0.25 h''_fg = h_fg + 0.80·c_pv·(T_s-T_sat)

Film Condensation (Nusselt Theory)

Vertical plate: h̄ = 0.943·(ρ_l(ρ_l-ρ_v)g·h'_fg·k_l³/(μ_l·L·(T_sat-T_s)))^0.25 h'_fg = h_fg + 0.68·c_pl·(T_sat-T_s) (modified latent heat)

Horizontal cylinder: h̄ = 0.725·(ρ_l(ρ_l-ρ_v)g·h'_fg·k_l³/(μ_l·D·(T_sat-T_s)))^0.25

Turbulent condensate (Re_f > 1800): h = k_l/L·(Re_f/4)^0.4·Pr_l^(-0.4)/(Re_f^0.5 - 5.82×10⁻²)

Output

Provide: Nu, h [W/m²K], q [W], temperature at key locations, correlation used and its validity range.