$(25$ points$)$ Consider a Rankine power cycle with $800C$ as the maximum temperature. The water $(10k\frac{g}{s})$ starts as a saturated liquid at state $1$ and undergoes the following processes: $12$ : Compression in a pump to a pressure of $3\mathrm{.}5$ MPa $(85\%$ isentropic efficiency$)$

$23$ : Heat addition at constant pressure to the maximum temperature

$34$ : Expansion in a turbine to the initial system pressure $(90\%$ isentropic efficiency$)$

$41$ : Heat rejection at constant pressure to the initial state.

The starting temperature will be varied from $20C$ to $200C$ to determine its effect on the cycle performance. Plot the following parameters on a graph as a function of this temperature $T_1$ :

The turbine exit quality $x_4($if superheated, change the value to $"1\mathrm{.}1"$ on the plot$)$

The net cycle power output

The cycle heat input

The cycle efficiency

The total exergy destruction rate in the cycle, assuming heat is rejected to $20C($the ambient$)$ and input at $800C$

Use EES to solve and submit formatted equations, parametric tables, and plots.

HINT $1$: Draw the T$-$s diagram if you are unsure of how to set up the equations.

HINT $2$: The total exergy destruction rate is tricky. The pump and turbine might be straightforward, but for the condenser and evaporator, carefully consider the temperatures at which heat transfers through. The temperatures in the devices themselves might not be constant, but their surrounding environment temperatures should be$.$