Chapter 5, Problem 5.96P

Special coatings are often formed by depositing thin layers of a molten material on a solid substrate. Solidification begins at the substrate surface and proceeds until the thickness S of the solid layer becomes equal to the thickness $\delta$ of the deposit.

1. Consider conditions for which molten material at its fusion temperature $T_f$ is deposited on a large substrate that is at an initial uniform temperature $T_i.$ With $S=0$ at $t=0,$ develop an expression for estimating the time $t_d$ required to completely solidify the deposit if it remains at $T_f$ throughout the solidification process. Express your result in terms of the substrate thermal conductivity and thermal diffusivity $\left(k_s,\alpha \right),$ the density and latent heat of fusion of the deposit $\left(\rho ,h_{sf}\right),$ the deposit thickness 6, and the relevant temperatures $\left(T_f,T_i\right).$
2. The plasma spray deposition process of Problem 5.25 is used to apply a thin $\left(\delta =2\text{mm}\right)$ alumina coating on a thick tungsten substrate. The substrate has a uniform initial temperature of $T_i=300\text{K,}$ and its thermal conductivity and thermal diffusivity may be approximated as $k_s=120\text{W/m}\cdot \text{K}$ and ${\alpha }_s=4.0\times {10}^{-5}{\text{m}}^{\text{2}}\text{/s,}$ respectively. density and latent heat of fusion of the alumina are $\rho =3970{\text{kg/m}}^{\text{3}}$ and $h_{sf}=3577\text{kJ/kg,}$ respectively, and the alumina solidifies at its fusion temperature $\left(T_f=2318\text{K}\right).$ Assuming that the molten layer is instantaneously deposited on the substrate, estimate the time required for the deposit to solidify.