The Efficiencies of Real Engines — Question 3

Question 3

Finite heat-exchanger temperature differences reduce the usable cycle span

External reservoirs are at 800 K and 300 K. A cyclic engine absorbs 1000 J at internal working temperature 700 K and rejects heat at 350 K. The engine core is reversible; irreversibility occurs only in the two thermal contacts.

1. Find core efficiency, work and rejected heat.

2. Calculate entropy generation in each contact and their sum. Compare with the maximum work between the external reservoirs.

Question 3 - Solution

Finite heat-exchanger temperature differences reduce the usable cycle span

(a) Method and calculation

$$\eta_{\mathrm{core}}=1-350/700=.500,\quad W=500\,\mathrm J,\quad Q_c=500\,\mathrm J.$$$$W_{\mathrm{external\ limit}}=1000(1-300/800)=625\,\mathrm J.$$

(b) Interpretation and further calculation

$$S_{{\mathrm{gen}},h}=1000(1/700-1/800)=5/28\,\mathrm{J/K}.$$$$S_{{\mathrm{gen}},c}=500(1/300-1/350)=5/21\,\mathrm{J/K}.$$$$S_{\mathrm{gen}}=5/12\,\mathrm{J/K},\quad625-500=300S_{\mathrm{gen}}=125\,\mathrm J.$$The external temperature difference is larger than the span available to the working cycle.

Physical check

The total entropy balance may use external reservoirs even when internal contact temperatures differ.