# 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 physcs.cup@gmail.com. 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.