Project ForgeTwo cylindrical shafts A and B at the same initial temperature are simultaneously placed in a furnace; their surfaces immediately reach furnace gas temperature. Shaft A: diameter 0.4 m, k=40 W/m-K, volumetric heat capacity 2×10⁶ J/m³-K. Shaft B: diameter 0.1 m, k=20 W/m-K, volumetric heat capacity 2×10⁷ J/m³-K. The centerline of shaft A reaches 400°C after 2 hours. The time (in hours) for the centerline of shaft B to reach 400°C is __________.
Explanation
With a fixed-surface-temperature boundary condition, the centerline temperature history depends only on the Fourier number Fo=αt/R², so equal dimensionless temperatures require equal Fo, i.e. t∝R²/α. α=k/(volumetric heat capacity): αA=40/2e6=2e-5 m²/s, αB=20/2e7=1e-6 m²/s. t_B/t_A=(R_B²/R_A²)(αA/αB)=(0.05/0.2)²×(2e-5/1e-6)=0.0625×20=1.25. t_B=1.25×2=2.5 hours.
Source: GATE ME 2016 Set-2 (2016)
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