Heat Transfer Mechanisms — Question 9

Question 9

Thermal resistance and heat capacity predict the time to a limit

A component of heat capacity $C=1200\,\mathrm{J/K}$ starts at ambient temperature 290 K. At $t=0$ a constant 20.0 W heater is switched on. The component loses heat to ambient through resistance $R=3.00\,\mathrm{K/W}$. Use $C\,dT/dt=P-(T-290)/R$, with uniform component temperature.

1. Derive $T(t)$, the steady temperature and the time to reach 340 K.

2. By that time, find energy supplied, energy stored and energy transferred to ambient. Would reducing $R$ to 2.00 K/W allow the temperature ever to reach 340 K under this model?

Question 9 - Solution

Thermal resistance and heat capacity predict the time to a limit

(a) Method and calculation

$$T(t)=290+PR(1-e^{-t/(RC)})=290+60(1-e^{-t/3600}).$$$$T_\infty=350\,\mathrm K,\quad 50=60(1-e^{-t/3600})\ \Rightarrow\ t=3600\ln6\approx6450.334\,\mathrm s.$$

(b) Interpretation and further calculation

$$E_{\mathrm{in}}=20t\approx129006.682\,\mathrm J,\quad\Delta U=C(340-290)=60000\,\mathrm J.$$$$Q_{\mathrm{out}}=E_{\mathrm{in}}-\Delta U\approx69006.682\,\mathrm J.$$$$R=2.00:\quad T_\infty=290+20(2)=330\,\mathrm K<340\,\mathrm K.$$The reduced-resistance component approaches 330 K monotonically and never reaches the stated limit.

Physical check

Input energy is not all stored: heat loss grows throughout the temperature rise.