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When water freezes in a pipe and the faucet is closed the water between the faucet and the frozen part cannot escape. This leads to increase in pressure as more water freezes. Because as long as there is still unfrozen water in the pipe the ice expands along the pipe. The increase in pressure is what causes a part of the pipe between the frozen part and the faucet to burst.

Now if we let the faucet trickle and assume the ice freezes in the same spot as when we do not let the faucet trickle the ice expanding from the frozen part will not be able to lead to a build up of pressure.

So apart from the fact that water freezes less likely when it is moving letting the faucet trickle should mainly be done to release the pressure between the faucet and the frozen part of the pipe. Note that ice can always expand in the other direction (to the water supply side) by displacing water.

Is this mechanism correct?

EDIT: Found this interesting video regarding pressure buildup

EDIT2 : After further reading, I came across the term latent heat. Because the evaporation of water cools the surrounding environment letting a faucet drip can actually accelerate freezing. But the freezing will then probably occurs first at the faucet itself. Given the above considerations, this could actually be beneficial in as such that there is no water trapped because the freezing starts at the faucet and gradually moves upstreams.

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    Not quite. Letting water trickle brings in water from the source that is presumably warmer, replacing the water trickling out preventing freezing. – George Anderson Feb 16 at 13:17
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    @GeorgeAnderson, the presumption is that the water freezes even with a trickle. That's certainly possible. See also any river in the north. The question is valid from a physics perspective, though maybe not from a home improvement one. There's no practical difference. – isherwood Feb 16 at 15:14
  • Get into it. – isherwood Feb 16 at 15:16
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    @isherwood Sure, agreed, that's why there are frozen waterfalls. I've had two bursted water pipes, they did a LOT of damage. My only point is that I believe it's a pretty common practice to leave faucets open to a trickle, not to just "keep water moving", but to replace it with presumably warmer water from the source. You need enough of a "trickle flow" to keep it from freezing, so I don't think we are disagreeing. – George Anderson Feb 16 at 16:00
  • Does this means that a toilet that likes to sporadically run won't suffer burst pipes? The only time that's good... They should invent a mechanism that would bleed valves under cold or high-pressure situations to prevent damage: a home BOP. – dandavis Feb 16 at 18:56
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Yes that is correct , the pipe generally burst where the water has not frozen. I once looked at burst tubes in a heat exchanger that was off-line. Individual exchanger tubes had 4 or 5 bursts each. The cold came in ( heat went out) through the baffles that were about two feet apart ; between the baffles there were bursts two feet apart where the water had not yet frozen.

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