Two plates of the same material and thickness L are at different initial temperatures T_i,1 and T_i,2, where _Ti,2 > T_i,1. Their faces are suddenly brought into contact. The external surfaces of the two plates are insulated.

(a) Let a dimensionless temperature be defined as T* (Fo) = (T – T_i,1)/( T_i,2 – T_i,1). Neglecting the thermal contact resistance at the interface between the
plates, what are the steady-state dimensionless temperatures of each of the two plates,T*_ss,1 and T*_ss,2? What is the dimensionless interface temperature T*_int at any time?

(b) An effective overall heat transfer coefficient between the two plates can be defined based on the instantaneous, spatially averaged dimensionless plate temperatures, U*_eff = q*/(T*₂ – T*₁). Nothing that a dimensionless heat transfer rate to or from either of the two plates may be expressed as q* = d(Q/Q_o), determine an expression for U*_eff for Fo > 0.2.