Superconductors — Question 9

Question 9

A lossless conductor can still require substantial refrigeration power

A cryogenic device has zero DC resistive loss but a constant heat leak $\dot Q_c=12.0$ W at $T_c=4.00$ K. Ambient temperature is $T_h=300$ K. The refrigerator achieves 10.0 percent of the Carnot coefficient of performance, $\mathrm{COP}_C=T_c/(T_h-T_c)$.

1. Find minimum reversible input power and actual refrigerator input power.

2. Compare actual input with a room-temperature alternative dissipating 6.00 kW, and find the largest heat leak that still uses less power than that alternative.

Question 9 - Solution

A lossless conductor can still require substantial refrigeration power

(a) Method and calculation

$$\mathrm{COP}_C=4/296=1/74,\quad W_{\min}=\dot Q_c/\mathrm{COP}_C=888\,\mathrm W.$$$$\mathrm{COP}=.10/74,\quad W_{\mathrm{actual}}=8880\,\mathrm W.$$

(b) Interpretation and further calculation

$$W_{\mathrm{actual}}/6000=1.48.$$$$\dot Q_c<6000(.10/74)=8.10811\,\mathrm W\quad\text{for a net power saving}.$$

Physical check

Zero electrical resistance does not imply zero system power; maintaining the cold environment has an energy cost.