A thermocouple has the following characteristics when it is immersed in a stirred bath: Mass of thermocouple =1 g Heat capacity of thermocouple =0.25cal/g∘C Heat transfer coefficient =20cal/cm 2 h∘C (for thermocouple and bath) Surface area of thermocouple =3 cm 2

A thermocouple has the following characteristics when it is immersed in a stirred bath: Mass of thermocouple $=1\mathrm{g}$ Heat capacity of thermocouple $=0.25\mathrm{cal}/\mathrm{g}\circ \mathrm{C}$ Heat transfer coefficient =20cal/cm2 h∘ C (for thermocouple and bath) Surface area of thermocouple =3 cm2 (a) Derive a transfer function model for the thermocouple relating the change in its indicated output $T$ to the change in the temperature of its surroundings Ts assuming uniform temperature (no gradients in the thermocouple bead), no conduction in the leads, constant physical properties, and conversion of the millivolt-level output directly to a ∘C reading by a very fast sensor. (b) If the thermocouple is initially out of the bath and at room temperature (23∘C), what is the maximum temperature that it will register if it is suddenly plunged into the bath (80∘C) and held there for $20\mathrm{s}$ ?