Work, Energy, and Emf — Question 10

Question 10

A thermal charge pump has an emf ceiling imposed by its heat engine

A heat engine operating between 600 K and 300 K drives a lossless electrical generator. It absorbs heat at 120 W and drives current 2.00 A. Ignore all dissipation except the thermodynamic efficiency limit.

1. Find the largest possible generator emf.

2. If the measured emf is 24.0 V, find engine efficiency and heat-rejection rate, and compare with the reversible limit.

Question 10 - Solution

A thermal charge pump has an emf ceiling imposed by its heat engine

(a) Method and calculation

$$\eta_C=1-T_c/T_h=1-300/600=.500.$$$$\mathcal E I\le\eta_C\dot Q_h\Rightarrow\mathcal E\le(.500)(120)/2.00=30.0\,\mathrm V.$$

(b) Interpretation and further calculation

$$P_e=\mathcal EI=48.0\,\mathrm W,\quad\eta=48.0/120=.400<.500.$$$$\dot Q_c=\dot Q_h-P_e=120-48=72.0\,\mathrm W.$$

Physical check

Even a lossless generator cannot overcome the efficiency limit of the engine supplying its mechanical work.