- 2070 Bhadra · 6 marks
Steam main with two insulation layers: overall coefficients
Short answer
After summing all thermal resistances for the 50 m length, the total resistance is 0.0294 K/W. The inside overall coefficient based on inner pipe area is U_i = 2.71 W/m²·K, and the outside overall coefficient based on outer insulation area is U_o = 1.08 W/m²·K.
A steam main of 8 cm inside diameter and 9.5 cm outside diameter is lagged with two successive layers of insulation. The layer in contact with pipe is 3.75 cm asbestos with thermal conductivity 0.11 W/m°K and the asbestos layer is covered with 1.5 cm thick magnesia insulation with thermal conductivity of 0.067 W/m°K. The inside film heat transfer co-efficient is 290 W/m²K and the outside film heat transfer co-efficient is 7.0 W/m²K. Conductivity of pipe material is 45 W/m°K. Calculate the inside and outside overall heat transfer co-efficient for 50 m length if the steam that is passing is at 350°C and the ambient air is at 30°C.
Given: r₁ = 4 cm = 0.04 m (inside), r₂ = 4.75 cm = 0.0475 m (outside of steel pipe), asbestos thickness 3.75 cm so r₃ = 0.085 m, magnesia thickness 1.5 cm so r₄ = 0.10 m. k_pipe = 45, k_asb = 0.11, k_mag = 0.067 W/m·K. h_i = 290, h_o = 7 W/m²·K. L = 50 m. T_steam = 350 °C, T_air = 30 °C (ΔT = 320 K).
Thermal resistances (for 50 m length)
| Resistance | Expression | Value (K/W) |
|---|---|---|
| Inside film | 1/(h_i 2π r₁ L) | 0.000274 |
| Pipe wall | ln(r₂/r₁)/(2π k L) | 0.000012 |
| Asbestos | ln(r₃/r₂)/(2π k L) | 0.016839 |
| Magnesia | ln(r₄/r₃)/(2π k L) | 0.007721 |
| Outside film | 1/(h_o 2π r₄ L) | 0.004547 |
| Total | 0.029394 |
Heat flow
Q = ΔT/R_total = 320/0.029394 = 10887 W (about 10.89 kW for 50 m; 218 W/m)
Overall heat transfer coefficients
Definition: Q = U A ΔT, so U = 1/(A R_total).
Inside area: A_i = 2π r₁ L = 2π × 0.04 × 50 = 12.566 m² U_i = 1/(A_i R_total) = 1/(12.566 × 0.029394) = 2.707 W/m²·K
Outside area: A_o = 2π r₄ L = 2π × 0.10 × 50 = 31.416 m² U_o = 1/(A_o R_total) = 1/(31.416 × 0.029394) = 1.083 W/m²·K
Check: U_i A_i = U_o A_o = 1/R_total = 34.02 W/K ✓.
✓ Answer checked by Claude Opus 5.5 (high effort)
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