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

Step 1: First order response.

\[ \theta(t) = \theta_{ss} - (\theta_{ss}-\theta_0)e^{-t/\tau} \]

Step 2: Find tau from initial rate.

\[ 2 = \frac{60}{\tau} \Rightarrow \tau = 30\ \text{s} \]

Step 3: Evaluate at t=60s.

\[ \theta(60) = 90 - 60\,e^{-2} = 90-8.12=81.88^\circ\mathrm{C} \]

\[ \boxed{\theta(60\ \mathrm{s}) \approx 81.9^\circ\mathrm{C}} \]