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Aluminium straight rectangular fin 3 × 50 × 100 mm in air: fin parameter, tip temperature, heat rate, efficiency, effectiveness and the Harper–Brown corrected length

ToolFins and Extended Surfaces
SourceHarper, D. R., and Brown, W. B. (1922). Mathematical Equations for Heat Conduction in the Fins of Air-Cooled Engines. NACA Report 158, eq. 5 (f′ = tanh(aw′)/aw′) and eqs. 31–32 (the corrected length w′ = w + t/2 against the exact value); Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A. S. (2007). Fundamentals of Heat and Mass Transfer, 6th ed. Wiley. Table 3.4 case A and Table 3.5.
LevelUndergraduate

Problem

A straight rectangular fin of pure aluminium (k = 237 W/(m·K)), 3 mm thick, 50 mm wide and 100 mm long, grows from a wall at 100 °C into air at 20 °C with h = 100 W/(m²·K) on all its surfaces including the tip. Find the fin parameter m and mL, the temperature at mid-length and at the tip, the heat rate, the fin efficiency and effectiveness, the fin resistance, and compare the exact convecting-tip solution with the Harper–Brown corrected-length approximation; check the Biot number for the one-dimensional theory.

Inputs to Enter

  • Fin type: straight fin, rectangular profile; tip condition: convection from the tip (Table 3.4 case A)
  • t = 3 mm, w = 50 mm, L = 100 mm; k = 237 W/(m·K), h = 100 W/(m²·K); T_b = 100 °C, T_∞ = 20 °C (SI mode)
Open the Fins and Extended Surfaces tool (loaded with this example)

Answers

  • P = 2(w + t) = 106 mm, A_c = 150 mm², m = √(hP/(kA_c)) = 17.27 /m, mL = 1.727, tanh mL = 0.9387; M = √(hPkA_c) θ_b = 49.11 W; h/(mk) = 0.02444
  • θ/θ_b = 0.4787 at mid-length (T = 58.3 °C) and 0.3371 at the tip (T_tip = 46.97 °C)
  • q_f = M [sinh mL + (h/mk) cosh mL]/[cosh mL + (h/mk) sinh mL] = 46.24 W; A_f = PL + A_c = 10,750 mm²; η_f = q_f/(hA_fθ_b) = 0.5376; ε_f = q_f/(hA_cθ_b) = 38.53; R_f = θ_b/q_f = 1.730 K/W
  • Harper–Brown: L_c = L + A_c/P = 101.42 mm, η = tanh(mL_c)/(mL_c) = 0.53764, q = M tanh(mL_c) = 46.24 W, within 6 × 10⁻⁷ of the exact case-A value (Report 158: below 0.1 % for an aluminium fin)
  • Bi = h(A_c/P)/k = 5.97 × 10⁻⁴ < 0.1: one-dimensional theory valid
  • USCS mode (0.1181 × 1.969 × 3.937 in, k 137 Btu/(h·ft·°F), h 17.61 Btu/(h·ft²·°F), 212 °F in 68 °F air): mL 1.726, q_f 157.8 Btu/h (= 46.24 W), η_f 0.5377

Notes

The tool's default inputs are this fin; press Submit. The case is the tool's own default checked against the closed forms with scipy (solve_bvp of the fin equation and a quad energy balance reproduce q_f to 1e-12; References/Fins/checks/fin_check.py) and against Report 158's eq. 32 example (a w′ = 1.836 gives f′ = 0.5176 against the exact 0.5149), in test_fin_efficiency.php (233 checks). Ten such fins on 200 cm² of bare base give η_o = 0.6102 and q_t = 622.4 W against 172 W without fins.