Question Two

$($a$)$ The temperature of soil is initially uniform at $20C.$ The soil surface is suddenly exposed to frigid winter temperature of $-15C$ for $60$ days. Determine the minimum depth, $x_m$ required for a water pipeline to be buried to avoid freezing. Soil properties are approximated as follows: density $=2050k\frac{g}{m^3},$ thermal conductivity $=0\mathrm{.}52\frac{W}{m}.K,$ heat capacity $=1840\frac{J}{k}g,$ thermal diffusivity $=0\mathrm{.}138{10}^{-6}\frac{m^2}{s}.$

$(6$ marks$)$