ShmoopTube

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And here's your shmoop du jour brought to you by sewer

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pipes which might be the most important part of our daily lives that we don't [Man sitting on a toilet and flushes]

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ever want to think about a sewer pipe carries stormwater out to the nearby lake

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over time sewage fields up inside ooh gross narrowing the opening in certain [sewage inside a sewer pipe]

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spots ooh double gross.. the city's chief sewage engineer makes several critical

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water flow measurements, in one spot a 20 centimeter thick layer of sludge has [sludge in the sewer pipe]

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narrowed to 2.0 meters diameter pipe take a look at this diagram right

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there yeah it shows that the sludge so we're going to call it politely goes

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completely around the pipe the water velocity at this point is 10 meters per

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second well based on this information what should the water velocity be when

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the sludge is cleared and here the potential answers.....So sewer sludge is

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really really gross and we're just going to move right ahead because we don't [Man slipping in sewer sludge]

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even want to think about it we've never been so happy to do math before.. Solve

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this well we'll use the fluid continuity equation that equation states that the

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product of the cross-sectional area of the pipe and the fluids velocity at one

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portion of a pipe equals the product of area and velocity at a second portion of

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the same pipe this is a function of the conservation of mass well basically if a [A garden hose pipe]

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volume X of fluid enters one portion of a pipe the same volume has to exit

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that same portion as a pipe narrows the velocity of the fluid increases as [Girl squirts boy with a water pipe]

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anyone who has ever put a thumb over the end of the garden hose can attest and if

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the pipe broadens well the velocity slows.. in this question we're not given [Dimensions of a pipe]

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the area but we're given the diameter and super genius types like us know that

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the area of a circle equals pi times one half of the diameter squared well we can

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plug that in on both sides of the equation and we can simplify it by

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cancelling out PI from both sides so the pipe was originally two meters but due [pipe's diameter measurements]

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to the twenty centimeters of sludge the diameter is reduced by forty centimeters

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since the layer of sludge encircles the pipe and now we're getting a little

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queasy from all this sludge talk, but we know that the current diameter is 1.6

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meters and since we know that the current velocity is 10 meters per second

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we plug those numbers in and we can solve for V sub 2 ...well V sub 2 equals V [Hand places V sub 2]

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sub 1 times d sub 1 squared over d sub 2 squared, which means it equals 10 meters

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per second times 1.6 meters squared over 2 meters squaresd, well simplifying the

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division helps us find that the post-sludge cleaning velocity is 6.4

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meters per second making B) the right answer... Now that we're finally done

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talking about sewer sludge we want to take a shower so we can feel clean again [Man runs away from sewer sludge and takes a shower]

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we'll try not to think about where all that water goes is it runs down the [water running down a drain]

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drain

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