Question

A thermometer initially at a steady temperature of \(30^\circ\mathrm{C}\) is inserted in a water bath maintained at \(90^\circ\mathrm{C}\). The initial rate of temperature rise of the thermometer is \(2\ ^\circ\mathrm{C\,s^{-1}}\). Assuming first order behavior, the thermometer reading (in \(^\circ\mathrm{C}\)) after one minute is ______ (rounded off to one decimal place).

Hint:

Use the initial slope \(d\theta/dt|_{t=0} = (\theta_{ss}-\theta_0)/\tau\) to find the time constant, then substitute \(t=60\ \mathrm{s}\) into \(\theta(t)=\theta_{ss}-(\theta_{ss}-\theta_0)e^{-t/\tau}\).

Answer 1

Correct Answer: 81.9

Using the deviation variable $y(t)=\theta_{ss}-\theta(t)$: $y(0)=60$, $\tau=30$ s, $y(60)=60e^{-2}=8.12$, $\theta(60)=90-8.12=81.88\approx81.9^\circ$C.