Consider a sheet of $20$ cm $($width$)20cm($length$)1cm($thickness$)$ with its top surface covered by a coating layer $($thickness of $2$ mm $).$ The water is flowing over the coating layer with a free stream temperature of $20C$ and velocity of $10\frac{m}{s}.$ All the other surfaces are well insulated, and the sheet and the coating have a uniform initial temperature of $95C($at $t=0).$

$($a$)$ At $t=0,$ determine the heat transfer rate from the coating surface to the water.

$($b$)$ Determine the Biot number of the sheet.

$($c$)$ How long does it take for the top surface of the sheet to be cooled to $30C?$

Properties of water: $v_w=4\mathrm{.}97{10}^{-4}\frac{m^2}{s},k_w=0\mathrm{.}65\frac{W}{m}*K,{}_w=0\mathrm{.}143{10}^{-6}\frac{m^2}{s},$ and $P_w=3\mathrm{.}15.$

Properties of the sheet: ${}_s=8400k\frac{g}{m^3},c_{p,s}=420\frac{J}{k}g*K,$ and $k_s=10\frac{W}{m}*K.$

operties of the coating: $k_c=0\mathrm{.}04\frac{W}{m}*K.$