A plastic coating is applied to wood panels by first depositing molten polymer on a panel and then cooling the surface of the polymer by subjecting it to airflow at \(25^{\circ} \mathrm{C}\). As first approximations, the heat of reaction associated with solidification of the polymer may be neglected and the polymer/wood interface may be assumed to be adiabatic.

If the thickness of the coating is \(L=1 \mathrm{~mm}\) and it has an initial uniform temperature of \(T_{i}=200^{\circ} \mathrm{C}\), how long will it take for the surface to achieve a safe-to-touch temperature of \(42^{\circ} \mathrm{C}\) if the convection coefficient is \(h=\) \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) ? What is the corresponding value of the interface temperature? The thermal conductivity and diffusivity of the plastic are \(k=0.25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=\) \(1.20 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\), respectively.

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L Air Th Plastic coating (k, a, T;) Wood panel