For this project, you will model a single cylinder going through a diesel cycle. This will be similar to the Example $9-3$ in your book, but with non$-$idealized compression and power strokes, and in metric units.

The Cycle

The diesel cycle to be considered begins with intake air at $T_1=25C$ and $P_1=100$kPa. The bottom$-$dead$-$center volume is $1920c{m}^3,$ or $A{A}_1=1\mathrm{.}920{10}^{-3}{m}^3.$

The compression stroke has an isentropic efficiency of ${}_{s,\text{c}\text{o}\text{m}\text{p}}=0\mathrm{.}80,$ and the expansion stroke has an isentropic efficiency of ${}_{s,\text{e}\text{x}\text{p}}=0\mathrm{.}70.$ The cold$-$air$-$standard can be used, which means that the air properties in the cycle are constant at the inlet conditions. This means: $R=0\mathrm{.}2870\frac{(kJ)}{(kg)(K)},c_p=1\mathrm{.}005\frac{(kJ)}{(kg)(K)},c_v=0\mathrm{.}718\frac{(kJ)}{(kg)(K)},$ and $k=1\mathrm{.}4.($Also$,$ air can be considered an ideal gas.$)$

The compression ratio of the cycle can be considered to be $18,$ which means $\frac{{}_1}{{}_2}=18.$ Begin your consideration with a cutoff ratio of $r_c=2,$ where $r_c=\frac{{}_3}{{}_2}.$

A Diesel cycle has two parts of the expansion stroke; compared to the ideal Diesel cycle the first part of the power stroke, $(2)(3),$ can still be considered isobaric, but the second one, $(3)(4),$ is no longer isentropic$.$ Recall that the isentropic efficiency of an expansion process is calculated as ${}_{s,\text{e}\text{x}\text{p}}=\frac{W_{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}}{W_s},$ where $W_s$ is the work produced by an isentropic process going through the same $P$:$W_{s,\text{e}\text{x}\text{p}}=m**(h_3-h_4)=m**c_p**(T_3-T_{4,s}),$ where State $(48)$ is characterized by a pressure of $P_4$ but an entropy of $s_{4s}=s_3,$ which means that an isentropic relation could be used for that point.

Note that for the isobaric process, $(2)(3),$ the heat input, $Q_{in},$ can be calculated as described in the book's example as the sum of the boundary work, $W_{23},$ and the $u$:$Q_{in}=m(h_3-h_2)=m{c}_p(T_3-T_2)$ and this is still valid. However, the book's example goes on to calculate $W_{\text{n}\text{e}\text{t}}=Q_{in}-Q_{\text{o}\text{u}\text{t}},$ and this is not valid, because of the losses in the compression and expansion strokes. You will have to instead calculate $u_{23}=m{c}_v{T}_{23}$ in addition to $Q_{23}=m{c}_p(T_3-T_2)$ to calculate the work produced in the isobaric part of the expansion stroke, $W_{23}.$

$1$

The Project

What you need to do for this project:

Compute the net work output, $W_{\text{n}\text{e}\text{t}}$ in a single stroke for this cycle, $W_{\text{n}\text{e}\text{t}}=W_{23}+W_{34}-W_{12}$

Calculate the heat input, $Q_{23},$ and heat rejected, $Q_{41}$ by the cycle

Calculate the thermal efficiency of the cycle, ${}_{th}.$

Modify the cutoff ratio, $r_c,$ to values between $1\mathrm{.}5$ and $5$ and plot the net work output, $W_{\text{n}\text{e}\text{t}}$ and thermal efficiency, ${}_{th}$ for a collection of values $($say: $1\mathrm{.}5,2,3,4,5).$ What can you say about the resulting plot?