An air-standard Diesel cycle has a compression ratio of 18 and a cut-off ratio of 2.2. At the beginning of the compression process, air is at 95 kPa and 28 ∘ C. Accounting for the variation of specific heats with temperature, determine (a) the volume at the beginning of the process.

Given values: - The initial temperature is $T_{1}=28^{\circ} \mathrm{C}$.

• The initial pressure is $P_{1}=95 \mathrm{kPa}$.

• The compression ratio is $r=18$.

• The cut-off ratio is $r_{c}=2.2$. Assumptions:

(1) The air standard assumptions are applicable.

(2) The kinetic and potential energy changes are negligible.

(3) Air is an ideal gas with variable specific heats.

The schematic diagram for the diesel cycle can be drawn as:

According to the ideal gas equation: $$ P_{1} V_{1}=m R T_{1} $$ Here, $R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ is the universal gas constant for air and mass is $m=1 \mathrm{~kg}$. On rearranging the above equation, we get, $$ V_{1}=\frac{m R T_{1}}{P_{1}} $$ Substitute the values in the above equation. $$ \begin{aligned} V_{1} &=\frac{(1 \mathrm{~kg})(287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K})\left[28^{\circ} \mathrm{C}+273\right] \mathrm{K}}{(95 \mathrm{kPa})\left(\frac{10^{3} \mathrm{~Pa}}{1 \mathrm{kPa}}\right)} \\ V_{1} & \approx 0.91 \mathrm{~m}^{3} \end{aligned} $$ Thus, the initial volume is $0.91 \mathrm{~m}^{3}$.