Efficiency of a 3 stage engine

A possible ideal-gas cycle operates as follows:

1. From an initial state ($p_1$, $V_1$) the gas is cooled at constant pressure to ($p_1$, $V_2$); Lets call the start and end temperature $T_1$ and $T_2$

2.The gas is heated at constant volume to ($p_2$, $V_2$);Lets call the start and end temperature $T_2$ and $T_3$

3.The gas expands adiabatically back to ($p_1$, $V_1$). Lets call the start and end temperature $T_3$ and $T_1$

Assuming constant heat capacities, show that the thermal efficiency η is

$$\eta=1-\gamma\frac{V_2/V_1 -1}{p_2/p_1-1}$$

Efficiencey is defined as: $$\eta=\frac{W}{Q_h}$$ the work done over the heat entered. The heat enters at stage 2 (and some leaves at stage 1 but that doesn't matter). So I need to find the heat entered at stage 2 and the work done.

Stage 1:

From the ideal gas equation we get: $$p_1V_1=nRT_1, \ \ \ \ p_2V_2=nRT_2 \implies \frac{T_2}{T_1}=\frac{V_2}{V_1}$$

The work done is just force times distance which is pressure times change in volume:

$$\Delta W=-p_1\Delta V=-p_1(V_2-V_1)$$

Stage 2:

It doesn't change in volume and so no work is done. However heat is put into the system, increasing the pressure. We need to find this heat.

$\Delta U= Q_h$

For an ideal gas we have: $$\Delta U= C_v\Delta T=C_v(T_3-T_2)$$

Where $C_v$ is heat capacity at constant volume.

Stage 3:

Stage 3 is adiabatic so $\Delta U=\Delta W=C_v(T_1-T_3)$

We also have, using the ideal gas law: $$T_3=\frac{p_2V_2}{p_1V_1}T_1$$

Let us sub this into the efficiency:

$$\eta=\frac{C_v(T_1-T_3)-p_1(V_2-V_1)}{C_v(T_3-T_2)}$$

If we get $T_3$ and $T_2$ in terms of T_1 and sub these in we get:

$$\eta=-1-\frac{p_1(V_2-V_1)}{C_vT_1(\frac{p_2V_2}{p_1V_1}-\frac{V_2}{V_1})}$$ And with the ideal gas law, with $n=1$ for simplicity we get $T_1=\frac{p_1V_1}{R}$

$$\implies\eta=-1-\frac{R(V_2/V_1-1)}{C_vT_1(\frac{p_2V_2}{p_1V_1}-\frac{V_2}{V_1})}$$

$R=C_p-C_v$ and $\gamma=C_p/C_v$

$$\implies\eta=-1-\frac{(\gamma-1)(V_2/V_1-1)}{C_vT_1(\frac{p_2}{p_1}-1)\frac{V_2}{V_1}}$$

I have no real clue really if this is right or wrong.

It's easiest to solve this problem in terms of the heats. For step 1, $$Q_1=\Delta H=nC_p(T_2-T_1)=nC_p\left(\frac{p_1V_2}{nR}-\frac{p_1V_1}{nR}\right)=\frac{C_p}{R}p_1(V_2-V_1)$$ For step 2, $$Q_2=nC_v(T_3-T_2)=nC_v\left(\frac{p_2V_2}{nR}-\frac{p_1V_2}{nR}\right)=\frac{C_v}{R}V_2(p_2-p_1)$$ For step 3, $$Q_3=0$$ Because this is a cycle, the total change in internal energy is zero. Therefore, the total work is equal to the sum of the three heats: $$W=Q_1+Q_2+Q_3=Q_1+Q_2$$ The heat added is just $Q_2$. Therefore, the efficiency is: $$\eta=\frac{W}{Q_2}=\frac{Q_1+Q_2}{Q_2}=1+\frac{Q_1}{Q_2}$$Substituting for $Q_1$ and $Q_2$ yields:$$\eta=1+\frac{C_p}{C_v}\frac{p_1(V_2-V_1)}{V_2(p_2-p_1)}=1-\gamma\frac{V_1/V_2-1}{p_2/p_1-1}$$