Problem $1$: Atkinson Cycle $(46$ points$)$

The ideal Atkinson cycle is a model for the combustion cycle used in hybrid$-$electric vehicles. The ideal Atkinson cycle consists of four processes:

$12$ : Adiabatic and isentropic compression.

$23$ : Constant volume heat addition.

$34$ : Adiabatic and isentropic expansion back to the initial pressure.

$41$ : Constant pressure heat rejection.

For this problem, at the beginning of the cycle, the air is at a pressure of $100$ kPa and a temperature of $300$ K $.$ The compression ratio is $11$ and the air temperature at the end of the heat addition process $($replacement for combustion$)$ is $1600$ K $.$

For this problem, carry out a cold air standard analysis of the Atkinson cycle. Consequently, you may assume that the ratio of specific heat is constants is independent of temperature and equal to $1\mathrm{.}4.$

To receive credit, you may not use thermofluids to solve this problem.

a$)$ Sketch the cycle in $p-v$ coordinates and sketch the corresponding Otto cycle with the same compression ratio. Indicate flows of work and$/$or heat for each process in the Atkinson cycle. $(11$ points$)$

b$)$ Populate a scorecard of the pressure $($ kPa $),$ temperature $),$ and specific volume $(\frac{m^3}{k}g)$ at each state in the cycle. $(12$ points$)$

c$)$ In an Otto cycle, the "expansion ratio" is the same as the compression ratio. What is the expansion ratio for the Atkinson cycle? $(3$ points$)$

d$)$ Express the cycle efficiency analytically in terms only of the temperature at each state and the ratio of the specific heats $(k).(10$ points$)$

e$)$ Calculate the cycle efficiency. How does this compare to a reversible cycle operating between the highest and lowest temperatures in the Atkinson cycle? $(5$ points$)$

f$)$ How does the cycle efficiency compare to a cold air standard Otto cycle with the same compression ratio? How could you have anticipated this result from the $p-v$ diagrams of the two cycles? $(5$ points$)$