Problem $2.(50$ pts$)$ Use the Laplace transform definition $($Equation $3-1$ in the textbook$)$ to find analytically

the Laplace transforms of the following function

$y(t)=5+t+e^{-2t}-cos0\mathrm{.}5t+t^2{e}^{-t}+e^{-at}sint$

Compare your solutions to the results obtained from the Laplace transform table. Problem $1.(35$ pts$)$ Consider the system shown in Figure $1.$ For tank $1,$ the steam is injected directly into

the liquid water, where it completely condenses into the liquid phase. Water vapor is only produced in the

second tank. $A_1$ and $A_2$ are the constant cross$-$sectional areas of the two tanks. $F_1,F_2$ and $F_3$ are the

volumetric flowrates and you can assume that the effluent flow rates are proportional to the liquid static

pressure that causes their flow, $F_2={}_1(h_1-h_2),F_3={}_2{h}_2.$ The heat transfer area for the steam coil is

$A_t$ and $U$ is the universal heat transfer coefficient between the superheated steam in the coil, $T_s$ is the

superheated steam temperature in the coil. ${}_{v_1}$ is the heat of condensation at the pressure of the first tank,

${}_{v_3}$ is the heat of vaporization at the pressure of the second tank, $T_{st,40}$ is the $40$ psi steam temperature, $C_{pv}$

is the $40$ psi steam heat capacity.

Water vapor

produced

Figure $1$

a$.(10$ pts$)$ Identify the state variables of the system.

b$.(5$ pts$)$ Determine what balances you should perform.

c$.(20$ pts$)$ Develop the state model that describes the dynamic behavior of the system.