$1)$ A $4$ cylinder engine runs at $3600$ rpm with a compression ratio of $5$ using the following cycle:

State $\backslash (1\backslash$rightarrow $\backslash )$ State $2$: Air at $100$ kPa and $300$ K is compressed isothermally from a volume of $1$ L down to its minimum volume. The initial entropy is $\backslash (5\mathrm{.}706[\backslash$mathrm$\{$kj$\}/\backslash$mathrm$\{$kg$\}-\backslash$mathrm$\{$k$\}]\backslash )($Not isentropic$)$

State $\backslash (\backslash$mathbf$\{2\}\backslash$rightarrow $\backslash )$ State $3$: The air is instantaneously heated to $700$ K

State $\backslash (3\backslash$rightarrow $\backslash )$ State $4$: The air is expanded isothermally back up to the bottom dead center $($Not isentropic$)$

State $\backslash (\backslash$mathbf$\{4\}\backslash$rightarrow $\backslash )$ State $1$: The air is cooled instantaneously back down to $300$ K

We are now stepping up the HW assignment quality. The standard is that someone not in the class could read your work and understand it$.$ Solve this question by hand. Use $\backslash ($ c$\_\{$v$\}=0\mathrm{.}7\backslash$mathrm$\{$~kJ$\}/\backslash$mathrm$\{$kg$\}-\backslash$mathrm$\{$K$\}\backslash ),\backslash ($ R$=0\mathrm{.}3\backslash$mathrm$\{$~kJ$\}/\backslash$mathrm$\{$kg$\}-\backslash$mathrm$\{$K$\}\backslash ),$ and $\backslash ($ k$=1\mathrm{.}4\backslash )$

a$)(3$ points$)$ Draw the system, find the mass in the system, as well as the minimum volume $\backslash (\backslash$mathrm$\{$dsa$\}(\backslash )$ top dead center$),$ and the maximum volume $($bottom dead center$).$

b$)(10$ points$)$ Sketch solution of one cycle in a P$-$V $("$Big$"$ volume$)$ and a T$-$s for guidance and identify the processes.

c$)(18$ points$)$ For each state show your control volume, and list all the state variables. $(\backslash (\backslash$mathrm$\{$P$\}\_\{1\}\backslash$mathrm$\{$~T$\}\_\{1\}\backslash )\backslash ($ s$\_\{1\}$ u$\_\{1\}$ V$\_\{1\}\backslash ))($Hint: "Big" V$,$ not specific volume$).$ And for each process show your energy equation, and list what stays constant between each state and its following state. $($ie: what stays constant between $\backslash (1\backslash$rightarrow $2,2\backslash$rightarrow $3\backslash$ldots $\backslash ))$

d$)$ Work and Heat

i$)(2$ points$)$ Find the isentropic work, and specific work from State $1$ to $2$

ii$)(2$ points$)$ Find the total and specific heat input from State $2$ to $3$

iii$)(2$ points$)$ Find the isentropic work, and specific work from State $3$ to $4$

iv$)(2$ points$)$ Find the total and specific heat output from State $4$ to $1$

v$)(2$ points$)$ Find the net work using the isothermal work out and work in to the system, and thermal efficiency $($Hint: use the isothermal work equation$)$

e$)(2$ points$)$ Using the net work you found in the previous question, what is the power output of the engine, in KW$?$ What is the mean effective pressure $($MEP$)?$