A large plate of thickness 2L is at a uniform temperature of T_i = 200°C, when it is suddenly quenched by dipping it in a liquid bath of temperature T∞ = 20°C. Heat transfer to the liquid is characterized by the convection coefficient h.

 

(a) If x = 0 corresponds to the midplane of the wall, on T - x coordinates, sketch the temperature distributions for the following conditions: initial condition (t ≤ 0), steady-state condition (t→∞), and two intermediate times.

(b) On - t coordinates, sketch the variation with time of the heat flux at x = L.

(c) If h - 100 W/m² ∙ K, what is the heat flux at x = L and t = 0? If the wall has a thermal conductivity of k = 50 W/m∙ K what is the corresponding temperature gradient at x = L?

Consider a plate of thickness 2L - 20 mm with a density of ρ - 2770 kg/m³ and a specific heat cρ = 875 J/kg ∙ K. By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate (J/m²) that is transferred to the bath over the time required to reach steady-state conditions.

(e) From other considerations, it is known that, during the quenching process, the heat flux at x = +L and x - -L decays exponentially with time according to the relation,  where t is in seconds,  Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process.