Entropy in the Real World: Refrigerators — Question 8

Question 8

Lowering a cold setpoint increases both the heat leak and work per joule

A refrigerated enclosure has heat leak $\dot Q_c=G(T_h-T_c)$, with $G=2.00$ W/K and $T_h=300$ K. Its actual coefficient is one-half the reversible value: $K_R=.500T_c/(T_h-T_c)$. Ignore all other loads.

1. Find cooling load and input power at $T_c=270$ K.

2. Repeat at 260 K and derive the input-power dependence on $T_c$. Explain the more-than-linear increase in required power.

Question 8 - Solution

Lowering a cold setpoint increases both the heat leak and work per joule

(a) Method and calculation

$$\dot Q_c=2(30)=60.0\,\mathrm W,\quad K_R=.500(270/30)=4.50.$$$$\dot W=60/4.50=13.3333\,\mathrm W.$$

(b) Interpretation and further calculation

$$\dot Q_c(260)=80.0\,\mathrm W,\quad K_R(260)=3.25,\quad\dot W(260)\approx24.6154\,\mathrm W.$$$$\dot W=\frac{G(T_h-T_c)^2}{.500T_c}.$$A larger temperature difference admits more heat and simultaneously raises the work cost of removing each joule.

Physical check

The coefficient model is stipulated for these two operating points; it is not a universal real-device law.