A very large copper ingot is removed from an annealing furna

A very large copper ingot is removed from an annealing furnace which has heated the ingot to a uniform temperature of 920 degree C. The ingot is then sprayed with high-pressure water jets, which have the effect of instantaneously cooling the outer surface of the ingot to 100 degree C Assuming the ingot is so large that you can presume that the error function equation represents the temperature distribution just below the cooled surface just after spraying commences, what is the temperature 10 mm below the surface after 10 second of spraying.

Solution

Thermal conductivity of copper is 398 W/mK.

Convert temperature to kelvin (0^oC = 273 K)

So heat removed per meter per second is thermal conductivity multiplied by temperature difference 326360 (W/m/s).

Multiplying heat removed by time 10 sec and distance 10 mm, dividing it thermal conductivity, we get temperature difference between the layer at 10 mm and inner temperature, which is equal to 75.8K

(1193 - T_10mm ) = 75.8

T_10mm = 1117.2K = 844.2^oC.