A liquid of initial temperature Ti is contained in a cylindrical vessel of radius R and height H, the walls of
which have negligible thermal capacity or resistance. The liquid is stirred with a magnetic stir bar, such that the internal heat transfer coefficient is h1. The vessel is insulated at the top surface, but receives heat at the bottom surface with a uniform flux q. Around the cylindrical surface, heat is being lost to a large bath at T∞, which is less than Ti , with a heat transfer coefficient h2. Develop the equation describing the bulk average temperature in the vessel T(t) using the lumped-capacity method. For the following specific case, determine the temperature in the vessel after 1 h. R = 10 cm; H = 20 cm; q = 1000 W/m2 ; h1 = 4000 W/m2 °C; h2 = 20 W/m2 °C; ρ = 900 kg/m3 ; ! C ˆ p = 4000 W·s/kg°C; Ti = 90°C; T∞ = 10°C, and determine the bulk average temperature in the vessel after one hour.