The Efficiencies of Real Engines — Question 10

Question 10

Internal irreversibility and auxiliary dissipation have separate work penalties

A cyclic engine takes 1000 J from 600 K and rejects heat to 300 K, generating .500 J/K internally. Its output drives auxiliaries consuming 50.0 J per cycle, all ultimately dissipated as heat at 300 K. The remaining work is delivered externally.

1. Find engine work, net work, total environmental heat rejection and total entropy generation.

2. Compare two upgrades: reducing internal entropy generation by .100 J/K, or reducing auxiliary consumption from 50.0 J to 20.0 J. Find net-work gain for each.

Question 10 - Solution

Internal irreversibility and auxiliary dissipation have separate work penalties

(a) Method and calculation

$$W_E=Q_h(1-T_c/T_h)-T_cS_{{\mathrm{gen}},E}=500-150=350\,\mathrm J.$$$$W_{\mathrm{net}}=350-50=300\,\mathrm J,\quad Q_{c,\mathrm{total}}=1000-300=700\,\mathrm J.$$$$S_{\mathrm{gen,total}}=700/300-1000/600=2/3\,\mathrm{J/K}=.500+50/300.$$

(b) Interpretation and further calculation

$$\Delta W_{\mathrm{internal}}=300(.100)=30.0\,\mathrm J.$$$$\Delta W_{\mathrm{auxiliary}}=50.0-20.0=30.0\,\mathrm J.$$Either upgrade alone raises net work to 330 J. Their equal benefit here follows from the supplied numbers, despite different physical loss mechanisms.

Physical check

Counting auxiliary dissipation inside both the engine entropy figure and the separate load would double-count it; the stated boundary avoids that.