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I ran into a bit of physics yesterday, and couldn't understand the physics at work. I've been scouring the Internet, but haven't found the answer yet. I'm guessing there's a basic principle I'm misinterpreting or ignorant of.

Basically, the sewer backed up into my house. Among other points of entry, water flowed up the shower drain, and up the toilet bowl.

The shower filled to a bit under the lip of its basin. The toilet filled to a bit under the lip of the bowl, but neither overflowed.

So basically I had a reservoir (the sewer), with one pipe (shower) extending to floor height + 4 inches, and another pipe (toilet) extending to floor height + 24 inches.

Where I'm getting confused is that in my physics textbooks I recall seeing a diagram of a U-shaped vessel with differently sized "legs." I think that the water level in each leg of the U had to be the same, because otherwise there'd be a pressure differential at the base of the U preventing the system from stabilizing.

Edit: Here's an example of what I'm talking about.

What would allow for this difference in water levels?

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The forces need to balance, and force is pressure*area. So the difference in pipe diameters and overflow areas (shower basin vs toilet bowl) must result in a balance of the force on the water. I'll let somebody work out a more full answer, currently studying! –  tpg2114 Oct 16 '13 at 22:18
    
@tpg2114 How does that differ from something like this? In that case the openings to the air are different diameters, should receive different forces from pressure, but remain at equal depth. –  AaronSieb Oct 16 '13 at 22:23
    
Yeah, tpg2114's answer probably isn't right. The water level should be the same at equilibrium. Or at least, it should be the same if the simple picture of them being tubes with open ends is correct. For the shower basin, this is probably true, but for the toilet, I'm not so sure (toilets have a raised water reservoir, and I'm not sure if this is connected with the bowl supply which supplies a pressure). Perhaps this would be more fit to ask at PlumbersExchange? –  DumpsterDoofus Oct 17 '13 at 2:31
    
@DumpsterDoofus Here's a diagram of a toilet. The tank isn't connected to the system in a meaningful way, although there could certainly be some bit of plumbing that matters further down. The toilet pipe DOES twist around a bit (it's usually a siphon), would that matter when the water level is above the "hump" in the pipe? –  AaronSieb Oct 17 '13 at 2:38

3 Answers 3

up vote 1 down vote accepted

What you're missing is some practical aspects of modern plumbing. Every modern plumbing fixture(toilet, shower/sink drain) has a trap or "u-bend" in the piping before it connects to the sewer branch or main pipe. This allows a pocket of water to sit in the bend of the pipe to keep sewer gases from escaping back into the room when the fixture is not being used. This water pocket obeys the equilibrium principle you know. When the fixture is used, the water above disturbs this balance and everything flows into the sewer pipe.

During your sewer back-up event, something has happened downstream to provide enough pressure reverse the normal flow, and the material will seek any free outlet (e.g. your shower drain). These events are usually very short-lived, so probably provided enough pressure to push stuff through the u-bend in the toilet and shower, then ceased. Once that pressure was gone, the normal physics applies and the liquid levels then reach equilibrium with their respective u-bends. Your shower basin, despite being lower than the toilet, contained the spill because it had a larger area to contain the volume of the spill.

Source: Plumbing design engineer.

More about plumbing traps here.

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To make a less apetizing point: The argument leading to the water levels being equal starts from an ideal fluid. Now a sewer backing up is far from ideal. I would expect solid pieces like toilet paper and feces clogging the pipe leading up to the shower (especially if the shower drain has a siphon), while the toilet drain is made for larger pieces fitting through.

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If you are concerned about the water height differences, then Every wash basin/toilet will have an over flow holes just around the lip of the toilet or just below the top surface of the wash basin. These tiny holes are not visible easily as it is made at places where it is not easily seen. This ensures the water level does not raise above certain level. That is why you not seeing overflowing. As you mentioned that these 2 pipes are at different height, the water in these two actually wanted to maintain same height, they don’t. Because these overflow holes doesn’t allows them to rise without completely filling these pockets which are inside the basin/toilet itself. There will be a lot of space before it again reaches to main lines. if you leave water for longer time, then you should be seeing it start filling these holes too and overflowing.

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I looked over the shower (which had the lowest water level), and it doesn't have overflow holes (or anywhere for the holes to hide). –  AaronSieb Oct 20 '13 at 15:31
    
won't the overflow spaces fill the same time as the main container? –  Andrey Dec 17 '13 at 15:41
    
@Andrey, no.the reason is because the very design itself.take a case of kitchen basin.the blockage mostly happen at the very opening(meshed) at he bottom.the over flow line will be joined just below this location which makes up an alternative flow path.if the block is somewhere in the middle of main pipe.then both the lines would be flooded. –  nemu Dec 18 '13 at 14:43
    
@nemu we are not talking about a block, we are talking about the sewer overflowing. All pipes connected to the sewer would in theory fill up to the same height. This would include any overflow paths that are only designed to help if the exhaust from the baison is blocked –  Andrey Dec 18 '13 at 14:49
    
You are right.if the sewer is blocked, then even overflow also gets filled.but slowly.the volume of space for overflow passages will be big.means it can accommodate more backing up water.but that too gets filled eventually. –  nemu Dec 18 '13 at 14:55

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