Question

A 4-mm-diameter and 10-cm-long aluminum fin (k = 237 W/m·K) is attached to a surface. If the heat transfer coefficient is 12 W/m2·K, determine the percent error in the rate of heat transfer from the fin when the infinitely long fin assumption is used instead of the adiabatic fin tip assumption.

Answer 1

The percent error in the rate of heat transfer from the fin when the infinitely long fin assumption is used instead of the adiabatic fin tip assumption is approximately 0.86%.

Calculate the heat transfer rates for both assumptions:

- Infinitely long fin assumption:

Q_inf = h * P * L * (T_b - T_inf)

- Adiabatic fin tip assumption:

m = √(h * P / (k * A_c))

θ = tanh(m * L)

Q_adiabatic = h * P * L * (T_b - T_inf) * (1 - θ / (m * L))

Determine the percent error:

Percent error = ((Q_inf - Q_adiabatic) / Q_inf) * 100%

Plug in the given values:

d = 4 mm = 0.004 m

L = 10 cm = 0.1 m

k = 237 W/m·K

h = 12 W/m²·K

Calculate the parameters:

A_c = π * (d/2)² = 1.2566 × 10⁻⁵ m²

P = π * d = 0.012566 m

m = √(12 * 0.012566 / (237 * 1.2566 × 10⁻⁵)) = 8.485 m⁻¹

θ = tanh(8.485 * 0.1) = 0.757

Calculate the heat transfer rates:

Q_inf = 12 * 0.012566 * 0.1 * (T_b - T_inf)

Q_adiabatic = 12 * 0.012566 * 0.1 * (T_b - T_inf) * (1 - 0.757 / (8.485 * 0.1))

Calculate the percent error:

Percent error = ((Q_inf - Q_adiabatic) / Q_inf) * 100% ≈ 0.86%