An exothermic process occurs uniformly through out a 10-cm-diameter sphere (k = 300 W/m · K, cp = 400 J/kg · K, ρ = 7500 kg/m3), and it generates heat at a constant rate of 1.2 MW/m3. The sphere initially is at a uniform temperature of 20°C, and the exothermic process is commenced at time t = 0. To keep the sphere temperature under control, it is submerged in a liquid bath maintained at 20°C. The heat transfer coefficient at the sphere surface is 250 W/m2 · K.
Due to the high thermal conductivity of sphere, the conductive resistance within the sphere can be neglected in comparison to the convective resistance at its surface. Accordingly, this unsteady state heat transfer situation could be analyzed as a lumped system.
(a) Show that the variation of sphere temperature T with time t can be expressed as dT/dt = 0.5 - 0.005T
(b) Predict the steady-state temperature of the sphere.
(C) Calculate the time needed for the sphere temperature to reach the average of its initial and final (steady) temperatures.
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