Answer by Rabbi Chaim Hillel Raskin – Rov in Kfar Chabad Beis, Israel:
When one releases water from such an urn, some water exits the tube and mixes into the main water chamber which is kept at a higher temperature. The question is whether doing so is considered cooking on Shabbos.
Halacha states that cold liquids—in contrast to dry solids—are subject to the Shabbos prohibition of cooking even after they were already completely cooked (yesh bishul achar bishul b’lach).
If, however, the liquid was completely cooked and is still appreciably warm (i.e. around 86° for illustrative purposes), heating it further is not prohibited—according to the Rema and Alter Rebbe—for it is viewed by halacha as “warming” as opposed to “cooking.”
While some poskim restrict one from bringing it to a boil, the Alter Rebbe implies that this is permissible.
Thus, if a pot of soup is on the blech where it is at least that temperature and all its contents are completely cooked, one may move it closer to the flame where it will begin to boil.
At what point are liquids considered fully cooked? Some poskim hold only once they were boiled; otherwise, even if they are very hot, one may not heat them further.
Others hold they are cooked once they have reached the heat of yad soledes bo (the upper bracket being 160°-174° F). The Alter Rebbe appears to side like this latter opinion, and so rule many contemporary halachic authorities.
When it comes to the urn, the water in the tube is estimated to be several degrees less than the primary water chamber. Yet, since that water definitely reached the heat of yad soledes when the urn initially boiled, and it is still at least warm, there is no concern when that water heats up more as it mixes with the hot water inside the urn.
Moreover, some claim that according to the laws of physics, hot and cold water that are touching will always mix between themselves, causing the heat to be distributed relatively evenly.
A halachic advantage of having this tube is so that one has advance warning before emptying the urn completely, thereby either causing the machine to shut off automatically or to burn out (and perhaps lead someone to add cold water).
Originally published in the Weekly Farbrengen, a newsletter by Merkaz Anash. See Sources