Problem $2$

Thermal System Modelling

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.

Consider the quenching process. If the thermal capacitance of the liquid bath is not large,

the heat energy transferred from the cube to the bath will change the bath temperature,

and we will need a model to describe the system. Consider the representation shown in

the above figure. The temperature of the cube is $T,$ the temperature of the bath is $T_b$ and

the temperature outside the bath is $T_o.$ The resistance between the cube and the bath is

$R1,$ and the resistance between the container wall and the liquid surface is R$2.$ The

capacitances of the cube and the liquid bath are $C$ and $C_b$ respectively.

Tasks:

Write the differential equations describing the system.

Write the State$-$Space Representation of the system.

Consider $T$ and $T_b$ to be the outputs of the system.

State$-$space equations:

$x=Ax+Bu$

$Y=Cx+Du$