Problem $5$

As the infrastructure engineer for the City of Troy the coming cold wave has made you nervous about the water main on $4^{th}$ street. The situation is shown Figure P$6\mathrm{.}4.$ Current temperature at the center of the main, $Tg,$ is $10C(50F).$ The projected average temperature for the cold wave, $T,$ is $-23C.$ A $1^{\text{'}\text{'}}(2\mathrm{.}54cm)$ 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 semi$-$infinite system with convection at the surface.

$\frac{T(y,t)-T_i}{T_{}-T_i}=[1-$erf$(\frac{y}{\sqrt[\phantom{2}]{4t}})]-[exp(\frac{hy}{k}+\frac{h^{2}t}{k^2})][1-$erf$(\frac{y}{\sqrt[\phantom{2}]{4t}}+\frac{h\sqrt[\phantom{2}]{t}}{k})]$

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?