Problem $5(39$ points$)$

$0\mathrm{.}5$ kilograms of water executes a Carnot power cycle. During the isothermal expansion, the water is heated from a saturated liquid to a saturated vapor at a pressure of $2$ MPa $.$ The vapor then expands adiabatically to a pressure of $10$ kPa where it has a quality of $74\mathrm{.}0\%.$ The vapor is then compressed isothermally until its quality is reduced to $24\mathrm{.}9\%.$

a$)$ Sketch the cycle on a $p-v$ diagram, including the vapor dome.

b$)$ Calculate the pressure, specific volume, and specific internal energy at each state.

c$)$ Calculate the heat, work, and change in internal energy for each process, in kJ $,$ and totals for the cycle. Do not invoke Qcycle $=$ Wcycle to solve the problem, but use this to check the validity of your solution.

d$)$ Calculate the cycle efficiency.

e$)$ Calculate the efficiency of a reversible heat engine operating between two reservoirs at this cycle's temperatures.

f$)$ Draw a cycle diagram showing the thermal reservoirs and the values of the flows of heat and cycle work.

g$)$ Evaluate the Clausius inequality for the cycle.