Question $3)$

$[3$ marks$]$

A simple thermal system consisting of two thermal capacitances $C_1$ and $C_2$ is shown in the figure. Thermal resistances $R_1$ separate the two capacitances from the ambient surroundings and thermal resistance $R_2$ is between the two capacitances. A heating device provides input heat flow $q_{in}$ to thermal chamber $1.$ The top and bottom boundaries of the system have perfect insulation $($infinite thermal resistance$).$

a$)$ Derive the complete mathematical model and Identify all dynamic variables and input variables. Use $q_{in}$ as the input and $T_1$ and $T_2$ as the outputs.

b$)$ Develop the SSR of the system.

c$)$ Use the following values for the system:

$C_1=5k\frac{J}{}C,C_2=50k\frac{J}{}C,T_2=25C,R_2=0\mathrm{.}05\frac{C}{W},R_1=0\mathrm{.}4\frac{C}{W}$

Assuming that initially $T_1$ and $T_2$ are equal to the ambient temperature and $q_{in}$ is about $2$ kW $,$ write a MATLAB code to estimate how long it takes for $T_2$ to reach a temperature of $60C.$ Include your MATLAB code and plots of $T_1$ and $T_2$ vs$.$ time.