Entropy in the Real World: Refrigerators — Question 1

Question 1

One device has different coefficients for cooling and heating

For a cyclic refrigerator, $Q_h=Q_c+W$, where $Q_c$ is heat removed from the cold region and $W$ is work input. Define $K_R=Q_c/W$ and heating coefficient $K_H=Q_h/W$. A device removes 900 J from a cold compartment while receiving 300 J of electrical work per cycle.

1. Find rejected heat and both coefficients of performance.

2. Derive the identity between $K_H$ and $K_R$. Explain why either coefficient may exceed one without violating energy conservation.

Question 1 - Solution

One device has different coefficients for cooling and heating

(a) Method and calculation

$$Q_h=900+300=1200\,\mathrm J,\quad K_R=900/300=3.00.$$$$K_H=1200/300=4.00.$$

(b) Interpretation and further calculation

$$K_H=\frac{Q_c+W}{W}=K_R+1.$$$$Q_h-W=Q_c>0.$$The delivered heating includes heat drawn from another region, so it need not be limited to the work input. These coefficients are heat-transfer ratios, not heat-engine efficiencies.

Physical check

Choose the desired output, cooling or heating, before defining the performance ratio.