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An industrial process contains the following step:


dirty water funneled through filter drips into clean water bin directly below

An upstream process dumps dirty water into the upper reservoir where it slowly drips through the filter, collecting in the lower reservoir. After the upper reservoir is completely empty, the lower reservoir is momentarily removed and poured into a downstream process.

Occasionally, the upper reservoir is overfilled with dirty water. In these cases, the system eventually reaches the following equilibrium state:

clean water level in bin is the same as dirty water level above filter

Notice there is still an amount X of dirty water remaining in the upper reservoir.

In this case, the lower reservoir cannot be removed because water will continue to filter through and drip on the equipment below. However, workers have devised an emergency solution: they manually siphon an appropriate amount of clean water out of the lower reservoir to make the necessary room.

Question: What is an "appropriate amount"? Is it amount X, or some other amount?

(More precisely: What is the minimum amount of clean water which must be removed from the lower reservoir in order to make enough room for all the dirty water in the upper reservoir to filter through, leaving the upper reservoir completely empty?)

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3 Answers 3

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The volume of water that has to be removed is (assuming vertical wall again) surface-to-tip times area of the surface. If we assume no dripping for a moment, the surface level will be below the tip, since the volume removed also includes the volume of the filter tube.

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Answer:

The workers need to siphon off a volume of water equal to the height of the remaining dirty water times the surface area of the lower container (assuming the walls are vertical at this point).

Reasoning:

After the siphoning, the situation will look like this: The situation after siphoning off water enough to have the filter dry The water level has been lowered by the height of the dirty water, over the entire surface of the container.

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    $\begingroup$ The nice thing about this is that it's very simple from the perspective of the engineers: just put the mouth of the siphon at a height level with the top of the filter. Regardless of the shape of the containers, the "right amount" is just however much until the water stops flowing. $\endgroup$ Commented May 26 at 19:15
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    $\begingroup$ Doesn't the filter contain water? Your solution doesn't completely empty the upper container as requested. $\endgroup$ Commented May 26 at 20:39
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Thanks for the answers.

Here's how I approached it.

The amount that needs to be siphoned off is equivalent to all the water (clean, dirty, and residing within the filter) above the dotted line:


solution_image

Notice that it is significantly more than amount X.

However, it is worth mentioning that not everything above the dotted line is water:

  1. There is the thickness of the walls of the upper reservoir.
  2. There is the filter itself (most likely packed with carbon particles or sand or ceramic or something). The water currently residing in the filter (i.e., the water in between the filter particles) needs to drain out, but the filter particles themselves will stay behind.
  3. And then there is a subtlety which is easy to miss. The dirty water presumably contains some amount of contaminants (that's what makes it dirty, after all). Those contaminants will remain behind in the filter. That is a very small volume which also must be subtracted.
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  • $\begingroup$ Why isn't the displaced volume of the upper container not considered? Lowering the level to the red line only takes out the water displaced by the submerged volume of the upper reservoir. If you removed the upper reservoir quickly so that most of the water still remains in the upper reservoir, the water level in the lower reservoir would immediately drop. $\endgroup$ Commented May 28 at 17:03
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    $\begingroup$ Thanks, @StevanV.Saban . I added a parenthetical to make it more clear. (BTW, nice mention of the Archimedes' principle in your answer!) $\endgroup$ Commented May 28 at 17:47

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