Question

An idealized diesel engine operates in a cycle known as air standard diesel cycle as shown. Fuel is sprayed into the cylinder at the point of maximum compression, $B$. Combustion occurs during expansion $BC$.

• A. Heat is released during process $DA$
• B. Heat is absorbed during process $BC$
• C. Heat is absorbed during process $CD$
• D. Efficiency of the cycle is
  $[1-\frac{1}{\gamma }\left(\frac{T_D-T_A}{T_C-T_B}\right)]$

Answer 1

Answer: (A), (B), (D)

Efficiency of the cycle is

$[1-\frac{1}{\gamma }\left(\frac{T_D-T_A}{T_C-T_B}\right)]$
For $A$ to $B$ : process is adiabatic so heat exchange is zero.

For $B$ to $C$: process is isobaric
So, $V\propto T$
$\Rightarrow V\uparrow ,T\uparrow$
Therefore, heat exchange is $Q_1=\Delta Q=n{C}_P\Delta T_{BC}$
$\Delta T=+ve$
So, heat is absorbed in process $BC$.
Hence, option $\left(A\right)$ is correct

For $C$ to $D$: process is adiabatic so heat exchange is zero.
Hence, option $\left(C\right)$ is incorrect

For $D$ to $A$: process is isochoric
$P\propto T$
$\Rightarrow P\downarrow ,T\downarrow$
Therefore, heat exchange is $Q_2=\Delta Q=n{C}_V\Delta T_{DA}$
$\Delta T_{DA}=-ve,$ so heat is released in process $DA$.
Hence, option $\left(B\right)$ is correct

Now, workdone is given as
$W=Q_1-Q_2$
Heat absorbed in one cycle is
$Q_{in}=Q_1$

Efficiency of cycle:
$\eta =\frac{\text{workdone}}{\text{net absorbed heat}}=1-\frac{Q_2}{Q_1}$
$=1-\frac{n{C}_V\Delta T_{DA}}{n{C}_P\Delta T_{BC}}$
$=1-\frac{1}{\left(\frac{C_P}{C_V}\right)}\left(\frac{T_D-T_A}{T_C-T_B}\right)$
$=1-\frac{1}{\gamma }\left(\frac{T_D-T_A}{T_C-T_B}\right)$
Hence, correct options are $\left(A\right),\left(C\right),\left(D\right)$