Problem B$-7-20$

Consider the thermal system shown in Figure $7-41.$ The temperature of the inflow liquid is kept constant, and the liquid inflow rate $Gk\frac{g}{s}$ is also kept constant. For the system is at steady state, wherein the heat input from the heater is $\frac{?}{\text{b}\text{a}\text{r}}(U)kca\frac{l}{s}$ and the temperature of the liquid in tank $2$ is $\frac{?}{\text{b}\text{a}\text{r}}({)}_{2}C.$ At $t=0,$ the heat input is changed from $\frac{?}{\text{b}\text{a}\text{r}}(U)$ to $\frac{?}{\text{b}\text{a}\text{r}}(U)+u,$ where $u$ is small. This change will cause the temperature of the liquid in tank $2$ to change from $\frac{?}{\text{b}\text{a}\text{r}}({)}_2$ to widehat$({)}_2+{}_2.$ Taking the change in the heat input from the heater to be the input to the system and the change in the temperature of the liquid in tank $2$ to be the output, obtain the transfer function ${}_2\frac{s}{U}(s).$ Assume that the thermal capacitances of tanks $1$ and $2$ are $C_1$kca$\frac{l}{}C$ and $C_2$kca$\frac{l}{}C,$ respectively, and that the specific heat of the liquid is ckca$\frac{l}{k}gC.$ The steady$-$state heat flow rates from tanks $1$ and $2$ are the same, $\frac{?}{\text{b}\text{a}\text{r}}(Q)kca\frac{l}{s}.$ The changes in heat flow rates from tanks $1$ and $2$ are $q_1,$ and $q_2,$ respectively.

Figure $7-41$ Thermal system.

Tank $2$