Physics Cup – TalTech 2024 – Problem 5

By Jaan Kalda (TalTech).

A water pipe of length L and of internal radius r runs under ground and is surrounded over its entire length by soil at temperature T_0. The specific heat of the soil is c, the density is \rho, and the heat conductance is \kappa. The characteristics of the pipe walls are identical to those of the soil. A boiler supplying water at a constant temperature T_1>T_0 is attached to the inlet of the pipe, and at the moment of time t=0, a tap is opened at the outlet of the pipe. The water in pipe starts flowing at a constant speed v. How long one has to wait for the water flowing out of the tap to become warm if it is known that this waiting time is significantly bigger than L/v? We call water warm if its temperature is higher than \frac 12(T_0+T_1). Only an estimate of the answer is required: you need to provide the correct functional dependence of the waiting time while estimating the magnitude of its parameters. The specific heat of water is c_w, and its density is \rho_w; the water flow in the pipe is turbulent so that the water temperature can be assumed to be constant over a cross-section of the pipe.

Please submit the solution of this problem via e-mail to First intermediate results and the first hint will be published on 21st April, 13:00 GMT. After the publication of the first hint, the base score is reduced to 0.9 pts. For full regulations, see the “Participate” tab.