Entropy in the Real World: Engines — Question 7

Question 7

A finite cold body warms as a reversible engine rejects heat to it

A reversible cyclic engine receives heat from a 600 K reservoir and rejects it to a finite body of constant heat capacity 500 J/K. The body warms from 300 K to 400 K. Assume heat is transferred reversibly at the instantaneous body temperature.

1. Find total rejected heat and the cold-body entropy increase.

2. Use entropy balance to find total hot heat input and work. Compare the overall efficiency with the efficiencies at the beginning and end.

Question 7 - Solution

A finite cold body warms as a reversible engine rejects heat to it

(a) Method and calculation

$$Q_c=C(400-300)=50000\,\mathrm J.$$$$\Delta S_c=C\ln(400/300)=500\ln(4/3)\approx143.841\,\mathrm{J/K}.$$

(b) Interpretation and further calculation

$$Q_h/600=\Delta S_c\ \Rightarrow\ Q_h\approx86304.6\,\mathrm J,\quad W=Q_h-Q_c\approx36304.6\,\mathrm J.$$$$\eta_{\mathrm{overall}}=1-\frac{100}{600\ln(4/3)}\approx.420657.$$$$\eta_{\mathrm{start}}=.500,\quad\eta_{\mathrm{end}}=1/3.$$The overall efficiency falls between its instantaneous endpoint values.

Physical check

Total rejected heat divided by the final sink temperature would underestimate the sink entropy increase.