Subtopic: Steady state heat transfer in conduction, convection and radiation
Question Statement:
A sphere (3.5 cm diameter) made of copper (ρ = 8954 kg m⁻³; Cp = 0.4 kJ kg⁻¹ K⁻¹; k = 375 W m⁻¹ K⁻¹) is initially at uniform temperature of 200 °C. It is suddenly placed in an environment of 35 °C having convective film coefficient of 12 W m⁻² K⁻¹. After 18 minutes of exposure, the temperature of the sphere in °C will be __________.
Official Answer Key:
123 to 124
Step-by-Step Solution & Explanation
• Check Biot number for lumped-capacitance validity: Lc = V/A = r/3 = 0.0175/3 = 0.00583 m. Bi = hLc/k = 12×0.00583/375 ≈ 0.00019 ≈ 0, so lumped-capacitance analysis applies.
• Lumped model: (T–T∞)/(Ti–T∞) = exp(–hAt/ρVCp) = exp(–h(A/V)t/(ρCp)); A/V = 3/r = 3/0.0175 = 171.43 m⁻¹.
• Time constant τ = ρCp/(h·A/V) = (8954×400)/(12×171.43) = 3,581,600/2057.1 ≈ 1741.3 s.
• t = 18 min = 1080 s. Exponent = –1080/1741.3 = –0.6203. exp(–0.6203) ≈ 0.5378.
• T(t) = 35 + (200–35)×0.5378 = 35 + 165×0.5378 = 35 + 88.7 ≈ 123.7 °C, inside the accepted range 123–124.
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