I am answering the following question:

Gas initially at a pressure of 101.325 kPa and temperature of 60 °C undergoes the following cycle:

  • 1-2: Adiabatic Compression through compression ratio of 4.5:1
  • 2-3: Heating at constant volume through a pressure ratio of 1.35:1
  • 3-4: Constant entropy expansion to initial pressure
  • 4-1: Constant pressure cooling to initial volume

If $Cp = 1\ \mathrm{\frac{kJ}{kg\ K}}$ and $Cv = 0.678\ > \mathrm{\frac{kJ}{kg\ K}}$ for the gas, determine the thermal efficiency for the cycle.

The given answer is 41.41%.

Since it sounds pretty much like the Otto cycle, I try using the formula with compression ratio $r_k = 4.5$:

$$k = Cp/Cv = 1.47$$

$$e = 1-(1/(r_k^{k-1})) = 50.68\% $$

What am I doing wrong? Is it not an Otto cycle?


What am I doing wrong? Is it not an Otto cycle?

No, it's not an Otto cycle. It would be if it wasn't for the last process, heat rejection (cooling) in Otto cycle is performed at constant volume not constant pressure.

This should be the $P-v$ diagram of your engine (sorry for the lousy image): enter image description here

So, naturally you can't apply Otto's efficiency to this cycle.

Side note: compression ratio in Otto cycle $r$ is not $\frac{P_2}{P_1}$ as in your calculations, it's defined as the ratio between maximum and minimum volume in the cycle $\frac{V_{max}}{V_{min}} = \frac{V_4}{V_3}$


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