A plate of thickness 2L, surface area A_s, mass M, and specific heat cp, initially at a uniform temperature T_i, is suddenly heated on both surfaces by a convection process (T_∞, h) for a period of time to, following which the plate is insulated. Assume that the midplane temperature does not reach T_∞ within this period of time.

 

(a) Assuming B_i ≫ 1 for the heating process, sketch and label, on T–x coordinates, the following temperature distributions: initial, steady-state (t →∞), T(x, t_o), and at two intermediate times between t to and t →∞.

(b) Sketch and label, on T – t coordinates, the midplane and exposed surface temperature distributions. (c) Repeat parts (a) and (b) assuming Bi ≪ 1 for the plate.

(d) Derive an expression for the steady-state temperature T(x, ∞) T_f, leaving your result in terms of plate parameters (M, c_p), thermal conditions (T_i, T_∞, h), the surface temperature T(L, t), and the heating time t_o.