Consider the water main of Example 5.6, which is buried in soil initially at 20°C and is suddenly subjected to a constant surface temperature of -15°C for 60 days. Use the Transient Conduction/Semi-Infinite Solid model of IHT to obtain the following solutions. Compare your results with those in the Comments section of the example.

(a) Calculate and plot the temperature history at the burial depth of 0.68 m for thermal diffusivities of a z 10⁷ = 1.0, 1.38, and 3.0m²/s.

(b) For a = 1.38 X 10^-7 m²/s, plot the temperature distribution over the depth 0 < x < 1.0 m for times of 1, 5, 10, 30, and 60 days.

(c) For a = 1.38 X 10^-7 m² /s, show that the heat flux from the soil decreases with increasing time by plotting q"_x(0, t) as a function of time for the 60-day period. On this graph, also plot the heat flux at the depth of the buried main, q"_x (0.68 m, t).