Chapter 2, Problem 2.68P

Typically, air is heated in a hair dryer by blowing it across a coiled wire through which an electric current is passed. Thermal energy is generated by electric resistance heating within the wire and is transferred by convection from the surface of the wire to the air. Consider conditions for which the wire is initially at room temperature $T_i,$ and resistance heating is concurrently initiated with airflow at $t=0.$

1. For a wire radius $r_o,$ an air temperature $T_{\infty },$ and a convection coefficient h, write the form of the heat equation and the boundary/initial conditions that govern the transient thermal response, $T\left(r,t\right),$of the wire.
2. If the length and radius of the wire are 500 mm and 1 mm, respectively, what is the volumetric rate of thermal energy generation for a power consumption of $P_{\text{elec}}=500\text{W?}$ What is the convection heat flux under steady-state conditions?
3. On $T-r$ coordinates, sketch the temperature distributions for the following conditions: initial condition $\left(t\le 0\right),$ steady-state condition $\left(t\to \infty \right),$ and for two intermediate times.
4. On $q_r^n-t$ coordinates, sketch the variation of the heat flux with time for locations at $r=0$ and $r=r_o.$