A copper sheet of thickness \(2 L=2 \mathrm{~mm}\) has an initial temperature of \(T_{i}=118^{\circ} \mathrm{C}\). It is suddenly quenched in liquid water, resulting in boiling at its two surfaces. For boiling, Newton's law of cooling is expressed as \(q^{\prime \prime}=\) \(h\left(T_{s}-T_{\text {sat }}\right)\), where \(T_{s}\) is the solid surface temperature and \(T_{\text {sat }}\) is the saturation temperature of the fluid (in this case \(T_{\text {sat }}=100^{\circ} \mathrm{C}\) ). The convection heat transfer coefficient may be expressed as \(h=1010 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{3}\left(T_{s}-T_{\text {sat }}\right)^{2}\). Determine the time needed for the sheet to reach a temperature of \(T=102^{\circ} \mathrm{C}\). Plot the copper temperature versus time for \(0 \leq t \leq 0.5\mathrm{~s}\). On the same graph, plot the copper temperature history assuming the heat transfer coefficient is constant, evaluated at the average copper temperature \(\bar{T}=110^{\circ} \mathrm{C}\). Assume lumped capacitance behavior.