As the infrastructure engineer for the City of Troy the coming cold wave has made you nervous about the water main on 4th street. The situation is shown Figure P6.35. Current temperature at the center of the main, \(T_{g \infty}\), is \(10^{\circ} \mathrm{C}\left(50^{\circ} \mathrm{F}\right)\). The projected average temperature for the cold wave, \(T_{\infty}\), is \(-23^{\circ} \mathrm{C}\). A \(1^{\prime \prime}(2.54 \mathrm{~cm})\) layer of packed snow covers the street.

a. What is the governing equation describing the temperature profile in the ground?

b. What are the boundary and initial conditions?

c. After looking in texts, you discover the following solution for conduction in a semiinfinite system with convection at the surface.

What will it take to adapt this solution to your problem, i.e., show the new formula?

d. Can the main survive a 3-day cold spell?

Transcribed Image Text:

T(x,t)-T T-T x == 401 hx hout exp + k k erf x 4aut + k