Exercise $2.$ To generate a water stream with a volumetric flow of $3\mathrm{.}0\frac{L}{m}in$ at $80\mathrm{.}0C,$ a tank with a heating

coil in a tank as shown in the following figure:

$-$Make a list of assumptions that includes parameter values needed for the energy balance.

$-$Propose a differential equation that describes the effect of the heat flow $(x)$ of the coil on the outlet

temperature $(Y).$

$-$Express the previous dynamic model in terms of the deviation variables and determine the constant

time, $,$ in minutes and the gain of the steady state, $K.$ Briefly explain the meaning of both.

$-$Determine the transfer function of the first order system and comment if the system is steady or not.

Explain your response.

$-$Represent graphically how $\frac{?}{\text{b}\text{a}\text{r}}(T)$ changes in the outlet stream when : $($i$)Q^{}=200W$ and $($ii$)$

$Q^{}=-200W.$

$-$Represent in one single graphic the inlet variable $x(t)$ and $Y\frac{t}{K}$ when the heating flow in the coil

increases at arate of $10W($i$.$e$.x(t)=10t).$ What does the time constant, $,$ represents in this

answer?