Chapter 10, Problem 2P

The initial temperature of the hot coffee cup shown in Fig. P10.2 is $T\left(0\right)=175°F$. The cup is placed in a room temperature of $T_{\infty }=70°F$. The temperature $T\left(t\right)$ of the coffee at time $t$ can be approximated by Newton’s law of cooling as

$\frac{dT}{dt}+kT\left(t\right)=h{T}_{\infty }$ where h is the convective heat transfer coefficient in $\frac{Btu}{f{t}^{2}h°F}$.

(a) Find the transient solution, $T_{tran}\left(t\right)$. What is the time constantof the response?

(b) Find the steady-state solution $T_{ss}\left(t\right)$.

(c) Determine the total solution $T\left(t\right)$.

(d) Sketch the total solution $T\left(t\right)$ .How long does it take for the temperature to reach 99% of the room temperature?

(e) Would increasing the value $h$ increase or decrease the time when the temperature of coffee reaches the room temp?

(f) Would a lower or higher value of $h$ be best for a cup of coffee?