Chapter 3, Problem 3.86P

An air heater may be fabricated by coiling Nichrome wire and passing air in cross flow over the wire. Consider a heater fabricated from wire of diameter $D=1\text{mm,}$ electrical resistivity ${\rho }_e={10}^{-6}\Omega \cdot \text{m,}$ thermal conductivity $k=25\text{W/m}\cdot \text{K,}$ and emissivity $\epsilon =\mathrm{0.20.}$ The heater is designed to deliver air at a temperature of $T_{\infty }=50°\text{C}$ under flow conditions that provide a convection coefficient of $h=250{\text{W/m}}^{\text{2}}\cdot \text{K}$ for the wire. The temperature of the housing that encloses the wire and through which the air flows is $T_{\text{sur}}=50°\text{C}\text{.}$

If the maximum allowable temperature of the wire is $T_{\max }=1200°\text{C,}$ what is the maximum allowable electric current 1? If the maximum available voltage is $\Delta E=110\text{V,}$ what is the corresponding length L of wire that may be used in the heater and the power rating of the heater? Hint: In your solution, assume negligible temperature variations within the wire, but after obtaining the desired results, assess the validity of this assumption.