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. 749
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. 749