Consider the following constant volume heating tank. The tank is perfectly mixed using a stirrer. The inlet cold water $)=(1000\frac{kg}{m^3}$ is heated using an electronic heater to the required design temperature.

Design Conditions:

Inlet temperature; $T_i=10C$

Outlet temperature; $T=50C$

Tank Volume; $V=0\mathrm{.}1m^3$

Heat capacity; $C_p=4\mathrm{.}0\frac{J}{g}.C$

Heater; electronic; $Q=80kW$

a$)$ Perform a dynamic energy balance and obtain the following form

$T_{(s)}^{\text{'}}=G_{d(s)}{T}_{i(s)}^{\text{'}}+G_{p(s)}{Q}_{(s)}^{\text{'}}$

b$)$ Is this system open$-$loop Stable? Explain why?

c$)$ Determine the tank temperature change time profile, if the inlet temperature is changed from $10C$ to $20C.$ Show a sketch $($with numbers$).$