Entropy in the Real World: Engines — Question 5

Question 5

A reversible engine cascade preserves the overall reservoir limit

Two reversible engines operate in series between 900 K, an intermediate temperature $T_m$, and 300 K. The first receives 900 J. All its rejected heat becomes input to the second, with no heat leak or accumulation at the intermediate stage.

1. For $T_m=600$ K, find each engine work and the overall efficiency.

2. Derive total work for arbitrary $T_m$. Find the intermediate temperatures that give equal stage works and equal stage efficiencies; are they the same?

Question 5 - Solution

A reversible engine cascade preserves the overall reservoir limit

(a) Method and calculation

$$Q_m=900(T_m/900)=600\,\mathrm J,\quad W_1=900-600=300\,\mathrm J.$$$$Q_c=600(300/600)=300\,\mathrm J,\quad W_2=300\,\mathrm J,\quad\eta_{\mathrm{overall}}=600/900=2/3.$$

(b) Interpretation and further calculation

$$W_1=900-T_m,\quad W_2=T_m-300\quad\text{in joules for these data}.$$$$W_1+W_2=600\,\mathrm J,\quad W_1=W_2\Rightarrow T_m=600\,\mathrm K.$$$$1-T_m/900=1-300/T_m\Rightarrow T_m=\sqrt{270000}\approx519.615\,\mathrm K.$$The intermediate temperature reallocates stage performance without exceeding the overall reversible limit.

Physical check

Overall efficiency is not the sum of stage efficiencies, since the second receives less heat.