Steel-reinforced concrete pillars are used in the construction of large buildings. Structural failure can occur at high temperatures due to a fire because of softening of the metal core. Consider a 200-mm-thick composite pillar consisting of a central steel core (50 mm thick) sandwiched between two 75-mm-thick concrete walls. The pillar is at a uniform initial temperature of T_i = 27°C and is suddenly exposed to combustion products at T_∞ = 900°C, h = 40 W/m²  · K on both exposed surfaces. The surroundings temperature is also 900C.

(a) Using an implicit finite difference method with Δx = 10 mm and Δt = 100 s, determine the temperature of the exposed concrete surface and the center of the steel plate at t = 10,000 s. Steel properties are: k_s = 55 W/m · K, p_s = 7850 kg/m³, and c_s = 450 J/kg · K. Concrete properties are: k_c = 1.4 W/m · K, p_c2300 kg/m³, c_c = 880 J/kg · K, and ε = 0.90. Plot the maximum and minimum concrete temperatures along with the maximum and minimum steel temperatures over the duration 0 ≤ t ≤ 10,000 s.

(b) Repeat part (a) but account for a thermal contact resistance of R"_t,c = 0.20 m² · K/W at the concretesteel interface.

(c) At t = 10,000 s, the fire is extinguished, and the surroundings and ambient temperatures return to T_∞ = T_sur = 27°C. Using the same convection heat transfer coefficient and emissivity as in parts (a) and (b), determine the maximum steel temperature and the critical time at which the maximum steel temperature occurs for cases with and without the contact resistance. Plot the concrete surface temperature, the concrete temperature adjacent to the steel, and the steel temperatures over the duration 10,000 ≤ t ≤ 20,000 s.