The Efficiencies of Real Engines — Question 4

Question 4

The ideal Otto model isolates the effect of compression ratio

An idealized Otto cycle has two reversible adiabatic legs and two constant-volume heat-transfer legs. States 1 to 2 compress by $r=V_1/V_2=8$; 2 to 3 heat at fixed volume; 3 to 4 expand back to $V_1$; 4 to 1 cool at fixed volume. Use constant $\gamma=1.40$, $T_1=300$ K and $T_3=1800$ K.

1. Find $T_2,T_4$ using $TV^{\gamma-1}=\text{constant}$ and derive the cycle efficiency from the two constant-volume heats.

2. Evaluate efficiency and compare it with the Carnot bound based on the extreme temperatures. Explain why the idealized cycle is not a prediction for a real engine.

Question 4 - Solution

The ideal Otto model isolates the effect of compression ratio

(a) Method and calculation

$$a=r^{\gamma-1}=8^{.4}\approx2.29740,\quad T_2=aT_1\approx689.219\,\mathrm K,\quad T_4=T_3/a\approx783.496\,\mathrm K.$$$$\eta=1-\frac{nC_V(T_4-T_1)}{nC_V(T_3-T_2)}=1-\frac1a.$$

(b) Interpretation and further calculation

$$\eta\approx.564725,\quad\eta_{\mathrm{Carnot,extremes}}=1-300/1800=5/6.$$$$.564725<.833333.$$Heat enters over varying gas temperatures. Real operation also departs from constant capacities, reversible compression and expansion, and the specified ideal heat-transfer model.

Physical check

The simplified efficiency depends on compression ratio only because its other model parameters are held fixed.