Hardness and other properties of metal can be improved by the rapid cooling that occurs during quenching, a process in which a heated object is placed into a liquid bath (see Figure below). The cube is immersed in an oil bath with temperature T, If the thermal capacitance of the liquid bath is not large, the heat energy transferred from the cube will change the bath temperature, and we will need a model to describe its dynamics. The temperature outside the bath is T., which is assumed to be known (ambient temperature). The convective resistance between the cube and the bath is R1, and the combined convective/conductive resistance of the container wall and the liquid surface is Ry. The capacitances of the cube and the liquid bath are C and C, respectively. Assume that T>T>T, Q1. Derive a model of the cube temperature, T, and the bath temperature Ty. (3pts) 02. Obtain the transfer function T(s)/T,(s) (3pts). 03. The thermal capacitance of the cube is C = 11.7 J/K and the thermal resistance is R, = 2.08 K.s/J. We suppose that T, is constant (= 100 °C) and Rz is infinite (R;- ), at which time the temperature of the cube will be equal to 1.98T, ?(Suppose that T(t-0) = 900°C). (4pts).
