Fins and Transient Conduction

Fins add surface area to push more heat into a fluid. Transient conduction asks how fast a body heats or cools; the simplest and most tested case is the lumped body.

Fins

A fin of perimeter P, cross-section A_c and conductivity k in a fluid with coefficient h has the fin parameter

m = √(hP/(kA_c))

With base excess temperature θ_b = T_b − T_∞:

A short fin with a convecting tip can be treated as an insulated fin with a corrected length L_c = L + A_c/P (L + t/2 for a thin rectangular fin, L + d/4 for a pin fin).

• Fin efficiency η_f = actual heat flow / heat flow if the whole fin were at base temperature. For an insulated tip: η_f = tanh(mL)/(mL).
• Fin effectiveness ε_f = heat flow with fin / heat flow from the base area without it. Fins are worth adding only if ε_f > 2 or so, which needs h small and k large (fins on the air side, not the water side).

Transient conduction: lumped capacitance

If the Biot number Bi = hL_c/k < 0.1 (L_c = V/A_s), the temperature inside the body is nearly uniform. Then:

(T − T_∞)/(T_i − T_∞) = exp(−hA_s t/(ρVc)) = exp(−Bi·Fo)

• Time constant τ = ρVc/(hA_s). After one time constant, the body has covered 63.2% of the temperature change.
• Fourier number Fo = αt/L_c² (α = k/(ρc), the thermal diffusivity).
• For a sphere L_c = r/3; for a long cylinder L_c = r/2.

For Bi > 0.1, use Heisler charts or the semi-infinite solid solution.

Worked example

A steel ball (r = 5 mm, ρ = 7800 kg/m³, c = 460 J/kg·K, k = 40 W/m·K) cools in air with h = 20 W/m²·K.

• L_c = r/3 = 1.667 mm; Bi = 20 × 0.001667/40 = 0.00083 → lumped is valid.
• τ = ρc L_c/h = 7800 × 460 × 0.001667/20 = 299 s.

Practice