The Efficiencies of Real Engines — Question 2

Question 2

A heat leak lowers efficiency even when the working engine is reversible

A source at 800 K supplies 1000 J per cycle to an installation rejecting heat to 400 K. Of the source heat, 200 J bypasses the engine and leaks directly to the cold reservoir. The remaining 800 J enters an internally reversible engine between the two reservoirs.

1. Find engine work, total rejected heat and installation efficiency.

2. Calculate total entropy generation and compare the work with the reversible limit for all 1000 J passing through the engine.

Question 2 - Solution

A heat leak lowers efficiency even when the working engine is reversible

(a) Method and calculation

$$W=(1000-200)(1-400/800)=400\,\mathrm J.$$$$Q_{c,\mathrm{total}}=400+200=600\,\mathrm J,\quad\eta_{\mathrm{installation}}=400/1000=.400.$$

(b) Interpretation and further calculation

$$S_{\mathrm{gen}}=600/400-1000/800=.250\,\mathrm{J/K}.$$$$W_{\mathrm{no\ leak}}=500\,\mathrm J,\quad\Delta W=100\,\mathrm J=400S_{\mathrm{gen}}.$$All entropy generation comes from bypass heat transfer; the internally reversible core does not make the installation reversible.

Physical check

Use total source heat in the installation efficiency denominator.