Problem 4. Recently, a team of Mechanical Engineers (Li et al., PNAS vol 110 (50

Question

Problem 4. Recently, a team of Mechanical Engineers (Li et al., PNAS vol 110 (50), 20023-20027, 2013) gave an argument that the huge stones that make up parts of China’s Forbidden City were transported along artificial ice paths lubricated with water. Part of the argument involves a calculation of the total amount of heat due to friction that diffuses into the ice-covered ground. They needed this to see if friction alone would maintain a ice-water interface or if the ancients needed to pour water onto the road to maintain the slick water-ice interface needed to transport the huge stones, 123 tons, which where reported to be transported 70 km in 28 days. They assumed that the rate at which heat diffusing into the ice-covered road per unit area is given by

dQ/ dt = k(T0 T1)/ sqrt(t)

where k, T0, T1, and of course are all constants, and t is the contact time.

If Q(0) = 0 and is the total contact time, compute how much heat per unit area diffused into the road, i.e., find Q( ) assuming Q(0) = 0.

Explanation / Answer

Problem 4. Recently, a team of Mechanical Engineers (Li et al., PNAS vol 110 (50

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