Project ForgeA steel ball of diameter 60 mm is initially in thermal equilibrium at 1030°C in a furnace. It is suddenly removed from the furnace and cooled in ambient air at 30°C, with convective heat transfer coefficient h=20 W/m²K. The thermo-physical properties of steel are: density ρ=7800 kg/m³, conductivity k=40 W/mK and specific heat c=600 J/kgK. The time required in seconds to cool the steel ball in air from 1030°C to 430°C is
- A519
- B931
- C1195
- 2144
Explanation
Characteristic length Lc=r/3=0.03/3=0.01m (sphere). Biot number Bi=hLc/k=20×0.01/40=0.005≪0.1, so lumped-capacitance is valid. Time constant τ=ρVc/(hA)=ρLc·c/h=7800×0.01×600/20=2340 s. θ/θ0=(T−T∞)/(T0−T∞)=(430−30)/(1030−30)=0.4. t=τ·ln(1/0.4)=2340×0.9163≈2144 s.
Source: GATE ME 2013 (Booklet Code A) (2013)
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