Temperature and Heat — Question 7

Question 7

A short contact pulse gives a temperature impulse

A small sensor has constant heat capacity $C=4.00\,\mathrm{J/K}$ and initially reads 290 K. An ideal thermal pulse has rate $P(t)=P_0\sin(\pi t/\tau)$ for $0\le t\le\tau$, with $P_0=20.0\,\mathrm W$ and $\tau=2.00\,\mathrm s$. Neglect every other transfer and use $C\,dT=P\,dt$.

1. Obtain $T(t)$ during the pulse and the final temperature.

2. When has the sensor received half the total pulse heat? Compare the true heat with a rectangular-pulse estimate $P_0\tau$.

Question 7 - Solution

A short contact pulse gives a temperature impulse

(a) Method and calculation

$$T(t)=290+\frac{P_0\tau}{\pi C}[1-\cos(\pi t/\tau)].$$$$Q_{\mathrm{total}}=\frac{2P_0\tau}{\pi}=25.465\,\mathrm J,\quad T_f=296.366\,\mathrm K.$$

(b) Interpretation and further calculation

$$1-\cos(\pi t/\tau)=1\ \Rightarrow\ t=\tau/2=1.00\,\mathrm s.$$$$Q_{\mathrm{rectangle}}=40.0\,\mathrm J,\quad\frac{Q_{\mathrm{rectangle}}}{Q_{\mathrm{total}}}=\frac\pi2\approx1.571.$$The rectangle overestimates by 14.535 J, or about 57.1% of the true heat.

Physical check

The temperature rises monotonically because the input rate never becomes negative during the pulse.