Entropy in the Real World: Engines — Question 2

Question 2

A target Carnot efficiency can be reached by changing either reservoir

A reversible engine operates between 600 K and 300 K. Use the entropy balance $Q_h/T_h=Q_c/T_c$ and the energy balance $W=Q_h-Q_c$.

1. Derive its efficiency and find the hot temperature needed for efficiency .600 while keeping the cold reservoir at 300 K.

2. Alternatively keep the hot reservoir at 600 K and find the required cold temperature. Compare the magnitudes of the two temperature changes.

Question 2 - Solution

A target Carnot efficiency can be reached by changing either reservoir

(a) Method and calculation

$$\eta=1-Q_c/Q_h=1-T_c/T_h.$$$$\eta_0=.500,\quad .600=1-300/T_h\ \Rightarrow\ T_h=750\,\mathrm K.$$

(b) Interpretation and further calculation

$$.600=1-T_c/600\ \Rightarrow\ T_c=240\,\mathrm K.$$$$\Delta T_h=+150\,\mathrm K,\quad\Delta T_c=-60\,\mathrm K.$$These are thermodynamic alternatives, not assertions of equal engineering difficulty. Ratios must use absolute temperatures.

Physical check

The sensitivity $\partial\eta/\partial T_c=-1/T_h$ differs from $\partial\eta/\partial T_h=T_c/T_h^2$.