$(20$ pt$)$ Consider the process shown in Figure P$8\mathrm{.}13$ in which steam with mass flowrate $M_{st}^{}$ is

condensing in the coil in order to heat liquid which is fed with mass flowrate $M^{}$ and temperature $T_0$ in

the well$-$stirred tank. The temperature is measured using a thermocouple that is placed in a

thermowell. Both the thermocouple and the thermowell exhibit first$-$order dynamics and the

mathematical model of the temperature dynamics and its measurement is

$M{C}_p\frac{dT}{dt}=M^{}{C}_p(T_0-T)+M_{st}^{}{H}_{\text{v}\text{a}\text{p}}$

${}_w\frac{d{T}_w}{dt}=T-T_w$

${}_m\frac{d{T}_m}{dt}=T_w-T_m$

Data:

$M=$ mass of the liquid in the tank $=100kg$

$C_p=$ specific heat capacity of the liquid $=1kJ*k{g}^{-1,C^{-1}}$

$T_0=$ temperature of the feed stream $=20C$

$T=$ temperature of the liquid in the tank $=60C($steady state value$)$

$M^{}=$ mass flowrate of the liquid $=10kg*mi{n}^{-1}$

$H_{\text{v}\text{a}\text{p}}=$ latent heat of vaporization of steam $=2000kJ*k{g}^{-1}$

${}_w=$ time constant of the thermowell $=1$min

${}_m=$ time constant of the thermocouple $=0\mathrm{.}1$min

Figure P$8\mathrm{.}13$

If the mass flowrate of the steam $M_{st}^{}$ is the input and the measured temperature $T_m$ is the output,

express the model in deviation variable form, after determining the steady state, and calculate

$($a$)$ the transfer function;

$($b$)$ the impulse response;

$($c$)$ the response to a step change of $M_{st}^{}$ by $+10\%$ of its steady$-$state value.

$($d$)($Bonus: $5$ pt$)$ Simulate the response to a sinusoidal input, with an amplitude of $10\%$ of steady$-$

state value and a period of $2$ minutes.

please solve it$,$ not just explaination