(Solved): Highly reflective aluminum coatings may be formed on the surface of a substrate by impacting the s ...

Highly reflective aluminum coatings may be formed on the surface of a substrate by impacting the surface with molten drops of aluminum. The droplets are discharged from an injector, proceed through an inert gas (helium), and must still be in a molten state at the time of impact. Consider conditions for which droplets with a diameter, velocity, and initial temperature of \( D=500 \mu \mathrm{m} \), \( V=3 \mathrm{~m} / \mathrm{s} \), and \( T_{i}=1100 \mathrm{~K} \), respectively, traverse a stagnant layer of atmospheric helium that is at a temperature of \( T_{\infty}=300 \mathrm{~K} \). What is the maximum allowable thickness, in mm, of the helium layer needed to ensure that the temperature of droplets impacting the substrate is greater than or equal to the aluminum coating surface temperature of \( 1000 \mathrm{~K} \). Properties of the molten aluminum may be approximated as \( \rho=2500 \mathrm{~kg} / \mathrm{m}^{3}, c=1200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \), and \( k=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} \).