A plane wal1 of thickness 0.6 m (L = 0.3 m) is made of steel (k = 30 W/m ∙ K. p = 7900 kg/m³, c = 640 J/kg ∙ K). It is initially at a uniform temperature and is then exposed to air on both surfaces. Consider two different convection conditions: natural convection, characterized by h = 10 W/m² ∙ K, and forced convection, with h = 100 W/m² ∙ K. You are to calculate the surface temperature at three different times-t = 2.5 min, 25 min, and 250 min-for a total of six different cases.

(a) For each of these six cases, calculate the non-dimensional surface temperature, θ^*: = (T_s – T_∞) / (T_i- T_∞), using four different methods: exact solution first-term-of-the-series solution, lumped capacitance, and semi-infinite solid. Present your results in a table.

(b) Briefly explain the conditions for which (i) the first-term solution is a good approximation to the exact solution, (ii) the lumped capacitance solution is a good approximation, (iii) the semi-infinite solid solution is a good approximation.