A furnace wall consists of 200 mm of firebrick (k = 0.8 W/m·K) and 50 mm of insulation (k = 0.05 W/m·K). The inner face of the brick is at 200 °C. The outer surface of the insulation loses heat by convection to air at 30 °C with h = 10 W/m²·K. Find the heat loss per m², the brick-insulation interface temperature and the outer surface temperature.
Show hints
Treat each layer and the convection film as thermal resistances in series, per unit area.
Show the explanation
Worked answer.
For one-dimensional steady conduction, each layer acts like an electrical resistor. Per square metre of wall:
Step 1: Thermal resistances (m²·K/W).
- Firebrick: R₁ = L/k = 0.2 / 0.8 = 0.25
- Insulation: R₂ = 0.05 / 0.05 = 1.00
- Outside convection: R₃ = 1/h = 1/10 = 0.10
- Total: R = 0.25 + 1.00 + 0.10 = 1.35 m²·K/W
Step 2: Heat flux. The overall temperature difference is 200 − 30 = 170 K.
q = ΔT / R = 170 / 1.35 ≈ 125.9 W/m².
Step 3: Interface temperature. The heat flux is the same through every layer.
T₂ = T₁ − qR₁ = 200 − 125.9 × 0.25 ≈ 200 − 31.5 = 168.5 °C.
Step 4: Outer surface temperature.
T₃ = T₂ − qR₂ = 168.5 − 125.9 × 1.00 ≈ 42.6 °C.
Check. The drop across the air film is q/h = 125.9/10 ≈ 12.6 K, and 30 + 12.6 = 42.6 °C. This matches Step 4, so the temperatures are consistent. Also, the total drop adds up: 31.5 + 125.9 + 12.6 = 170 K.
Interpretation. The 50 mm insulation layer provides about 74% of the total resistance (1.00 of 1.35), and most of the temperature drop occurs across it. This is why thin layers of low-conductivity insulation are so effective. Without insulation, R = 0.35 and q = 170/0.35 ≈ 486 W/m², nearly four times the loss.
Assumption to state. The interface temperature of 168.5 °C must be within the insulation's rated service temperature; otherwise a different insulation material is needed. We also ignored radiation from the outer surface and contact resistance between layers.
Common mistake. Adding conductivities instead of resistances, or forgetting the convection resistance and assuming the outer surface is at air temperature. Without the film, q would be 170/1.25 = 136 W/m², an overestimate.
Likely follow-up. "Does adding insulation always reduce heat loss?" For flat walls, yes. For small pipes and wires, adding insulation can increase heat loss until the critical radius r_c = k/h is exceeded.
Original UniCircle question. Last reviewed 28 Oct 2026.