$5($b$)$ Fig. $13$ shows a simple model of an industrial furnace. A packing of temperature $T_1$ is being heated in the furnace by an electric heater supplying heat at the rate $Q_{hi}(t).$ The temperature inside the furnace is $T_2,$ the walls are at temperature $T_3,$ and ambient temperature is $T_a.$ The thermal capacitances of the packing, the air inside the furnace, and the furnace walls are $C_{h1},C_{h2},$ and $C_{h3},$ respectively. Derive state$-$variable equations for this system assuming that heat is transferred by convection only, with the convective heat transfer coefficients $h_{cl}($air packing$),h_{c2}($air inside walls$),$ and $h_{c3}($outside walls$-$ambient air$).$

MPC$6/1-1$

Fig. $13$ Simplified model of an industrial furnace.