Determine an expression for the depth below the surface of a semi-infinite solid at which the rate of cooling is maximum. Substitute the information given in Problem 18.22 to estimate how far below Earth’s surface this maximum cooling rate is achieved.

Data From Problem 18.22

In the canning process, sealed cans of food are sterilized with pressurized steam in order to kill any microorganisms initially present in the food and thereby prolong the shelf life of the food. A cylindrical can of food has a diameter of 4 cm and length of 10 cm. The food material has a heat capacity of 4000 J/kg · K, a density of 1500 kg/m³, and thermal conductivity of 0.6 W/m · K. In the present process, steam, at 120°C, is used to sterilize the can. The convective heat-transfer coefficient is 60 W/m² · K. Initially, the can and its contents are at a uniform temperature of 20°C.