The Second Law of Thermodynamics — Question 5

Question 5

Coupled devices can be checked by canceling their internal work exchange

An engine between 600 K and 300 K absorbs 1000 J and produces 400 J work. It drives a refrigerator between the same reservoirs. The refrigerator absorbs 300 J from the cold reservoir using all 400 J work. Both devices are cyclic.

1. Find the two rejected heats and each device entropy generation.

2. Cancel the internal work transfer and find the net reservoir heat transfers and total entropy generation of the composite.

Question 5 - Solution

Coupled devices can be checked by canceling their internal work exchange

(a) Method and calculation

$$Q_{c,E}=1000-400=600\,\mathrm J,\quad Q_{h,R}=300+400=700\,\mathrm J.$$$$S_{{\mathrm{gen}},E}=600/300-1000/600=1/3\,\mathrm{J/K}.$$$$S_{{\mathrm{gen}},R}=700/600-300/300=1/6\,\mathrm{J/K}.$$

(b) Interpretation and further calculation

$$W_{\mathrm{external}}=0,\quad Q_{h,\mathrm{net\ out}}=1000-700=300\,\mathrm J.$$$$Q_{c,\mathrm{net\ in}}=600-300=300\,\mathrm J,\quad S_{\mathrm{gen,total}}=300/300-300/600=.500\,\mathrm{J/K}.$$The composite simply transfers heat downhill while producing no external work.

Physical check

Internal exchanges cancel in the composite balance, but entropy generation adds.