1D conduction through insulation: An insulating wall has thermal conductivity K = 0.04 W/m·K and thickness L = 0.16 m. The steady heat flux from outside to inside is 10 W/m^2, and the inside surface is at −5 °C. What is the outside surface temperature?

Introduction / Context:
One-dimensional steady conduction through a plane wall relates heat flux, material conductivity, thickness, and the surface temperature difference. This problem applies Fourier's law to find the unknown outside surface temperature given heat flux and inside temperature.

Given Data / Assumptions:

• K = 0.04 W/m·K (thermal conductivity).
• L = 0.16 m (wall thickness).
• q = 10 W/m^2 (heat flux from outside to inside).
• T_inside = −5 °C; outside temperature T_outside = ?
• Steady state, 1D conduction, no internal heat generation.

Concept / Approach:
Fourier's law for a plane wall: q = K * (T_out − T_in) / L when heat flows from outside to inside. Rearranging allows solving for the unknown T_out when q, K, L, and T_in are known.

Step-by-Step Solution:
Start with q = K * (T_out − T_in) / L.Insert values: 10 = 0.04 * (T_out − (−5)) / 0.16.Compute K/L: 0.04 / 0.16 = 0.25.Solve: 10 = 0.25 * (T_out + 5) ⇒ T_out + 5 = 40 ⇒ T_out = 35 °C.

Verification / Alternative check:
Back-calc flux with T_out = 35 °C: q = 0.04 * (35 − (−5)) / 0.16 = 0.04 * 40 / 0.16 = 10 W/m^2, matching the given value.

Why Other Options Are Wrong:
25 °C, 30 °C, 40 °C yield fluxes of 7.5, 8.75, and 11.25 W/m^2 respectively with the same parameters, not equal to 10 W/m^2.

Common Pitfalls:

• Sign mistakes in temperature difference when one side is below 0 °C.
• Using thermal resistance incorrectly (R = L/K) without consistent units.

Final Answer:
35 °C