(Solved): A long, cylindrical, electrical heating element of diameter D=10mm, thermal conductivity k=240W ...

A long, cylindrical, electrical heating element of diameter $D=10\mathrm{mm}$, thermal conductivity $k=240\mathrm{W}/\mathrm{m}?\mathrm{K}$, density $?=2700$ $\mathrm{kg}/{\mathrm{m}}^3$, and specific heat $c_p=900\mathrm{J}/\mathrm{kg}?\mathrm{K}$ is installed in a duct for which air moves in cross flow over the heater at a temperature and velocity of $27^{?}\mathrm{C}$ and $20\mathrm{m}/\mathrm{s}$, respectively. (a) Neglecting radiation, estimate the steady-state surface temperature when, per unit length of the heater, electrical energy is being dissipated at a rate of $1000\mathrm{W}/\mathrm{m}$. (b) If the heater is activated from an initial temperature of $27^{?}\mathrm{C}$, estimate the time required for the surface temperature to come within $10^{?}\mathrm{C}$ of its steady-state value. Evaluate the properties of air at $450\mathrm{K}$.