Project ForgeA metal ball of diameter 60 mm is initially at 220°C. The ball is suddenly cooled by an air jet of 20°C. The heat transfer coefficient is 200 W/m²·K. The specific heat, thermal conductivity and density of the metal ball are 400 J/kg·K, 400 W/m·K and 9000 kg/m³, respectively. The ball temperature (in °C) after 90 seconds will be approximately
- 141
- B163
- C189
- D210
Explanation
For a sphere of radius r=0.03 m, characteristic length V/A = r/3 = 0.01 m. Biot number Bi = h(V/A)/k = 200×0.01/400 = 0.005 ≪ 0.1, so the lumped-capacitance method applies. Time constant τ = ρc(V/A)/h = 9000×400×0.01/200 = 180 s. θ/θ0 = e^(−t/τ) = e^(−90/180) = e^(−0.5) ≈ 0.6065. T = 20 + (220−20)×0.6065 ≈ 20+121.3 ≈ 141.3°C.
Source: GATE ME 2017 Session 2 (ME-2) (2017)
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