$($a$)$ Suppose you invented a cyclic machine that generates work and runs between two thermal reservoirs of temperatures $287C$ and $27C.$

Illustrate this process on a fully labelled system diagram.

Calculate the highest possible efficiency for this machine.

Another inventor claims he has made a similar machine $($i$.$e$.$ running between two thermal reservoirs of the same temperatures of $287C$ and $27C)$ but with an efficiency of $60\%.$ Comment on feasibility of his claim.

$[7$ marks$]$

$($b$)$ Consider the reversible adiabatic flow of steam through a nozzle. Steam enters the nozzle at $5$ MPa and $400C,$ with a velocity of $30\frac{m}{s}.$ The pressure of the steam at the nozzle exit is $0\mathrm{.}5$ MPa $.$ Determine the exit velocity of the steam from the nozzle, assuming a reversible, adiabatic, steady$-$state process.

$[9$ marks$]$

$($c$)$ Show, using a system diagram, how the heat pump and heat engine may be coupled together through the shaft work. If the efficiency of the engine is $45\%,$ and the coefficient of performance of the heat pump is $2\mathrm{.}8$ :

i$)$ Calculate the overall efficiency of the system.

$[2$ marks$]$

ii$)$ Compare and comment on the efficiency of the overall system to heat the building directly with heat from the high temperature reservoir.

$[3$ marks$]$

$($d$)150$ J of heat flows from a hot reservoir at $700$ K to a cold reservoir at $350$ K $.$

i$)$ Calculate the net entropy change of this system.

$[2$ marks$]$

ii$)$ Discuss the effect of reducing the temperature of the hot reservoir to $600$ K $,$ but still allowing $150$ J to flow into the cold reservoir at $300$ K in relation to this entropy change?

$[2$ marks$]$