3. Response of Thermometer

Heat balance: \[hA(T_\infty-T) - 0 = mC_P\frac{dT}{dt}\] Rearranging the above, \[\begin{aligned} \frac{mC_P}{hA}\frac{dT}{dt} + T &= T_\infty \\ \tau\frac{dT}{dt} + T &= T_\infty \\ \text{At steady state, i.e., for $t<0$} 0 + T_s &= T_{\infty s}\\ \text{Using deviation variables, $\theta=T-T_s; \ \theta_\infty=T_\infty- T_{\infty s}$, we get} \\  \tau \frac{d\theta}{dt} + \theta &= \theta_{\infty} \end{aligned}\] \\ Taking Laplace transform, and rearranging \[\boxed{\frac{\theta(s)}{\theta_\infty(s)} = \frac{1}{\tau s+1}}\]