A large plate of thickness 2L is at a uniform temperature of T; = 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 mid plane 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  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 = O? If the wall has a thermal conductivity of k = 50 W/m' K, what is the corresponding temperature gradient at x = L?

(d) Consider a plate of thickness 2L = 20 mm with a density of p = 2770 kg/m³ and a specific heat c_p = 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, q" = A exp (- Bt), where t is in seconds, A = 1.80 X 10⁴ W/m² ∙ and B = 4.126 X 10^{- 3}s ^-1. Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process.

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