A constant-property, one-dimensional plane wall of width 2L, at an initial uniform temperature T_i, is heated convectively (both surfaces) with an ambient fluid at T_∞ = T_∞ ,1, h = h₁. At a later instant in time, t = t₁, heating is curtailed, and convective cooling is initiated. Cooling conditions are characterized by T_∞ = T_∞,2 = T_i, h = h₂.

(a) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the heating phase (Phase 1). Express the equations in terms of the dimensionless quantities θ*, x*, Bi₁, and Fo, where Bi₁ is expressed in terms of h₁.

(b) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the cooling phase (Phase 2). Express the equations in terms of the dimensionless quantities θ*, x*, Bi₂, Fo₁, and Fo where Fo₁ is the dimensionless time associated with t₁, and Bi₂ is expressed in terms of h₂. To be consistent with part (a), express the dimensionless temperature in terms of T_∞ = T_∞,1.

(c) Consider a case for which Bi₁ = 10, Bi₂ = 1, and Fo₁ = 0.1. Using a finite-difference method with Δx* = 0.1 and ΔFo = 0.001, determine the transient thermal response of the surface (x* = 1), midplane (x* = 0), and quarter-plane (x* = 0.5) of the slab. Plot these three dimensionless temperatures as a function of dimensionless time over the range 0 ≤ Fo ≤ 0.5.

(d) Determine the minimum dimensionless temperature at the midplane of the wall, and the dimensionless time at which this minimum temperature is achieved.