The roof of a house consists of a 15-cm-thick concrete slab (k = 1.4 W/m · K and α = 0.69 × 10^-6 m²/s) that is 18 m wide and 32 m long. One evening at 6 pm, the slab is observed 

to be at a uniform temperature of 18°C. The average ambient air and the night sky temperatures for the entire night are predicted
to be 6°C and 260 K, respectively. The convection heat transfer coefficients at the inner and outer surfaces of the roof can be taken to be h_i = 5 and h_o = 12 W/m² · K, respectively. The house and the interior surfaces of the walls and the floor are maintained at a constant temperature of 20°C during the night, and the emissivity of both surfaces of the concrete roof is 0.9. Considering both radiation and convection heat transfers and using the explicit finite difference method with a time step of Δt = 5 min and a mesh size of Δx = 3 cm, determine the temperatures of the inner and outer surfaces of the roof at 6 am. Also, determine the average rate of heat transfer through the roof during that night.

Transcribed Image Text:

Radiation & Radiation T₁ Tsky Convection hos To h₁, Ti Concrete roof Convection 15 cm