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Toilets

                         

Toilets are everywhere, in our parks, our schools, and even in our homes. Whether or not you'd like to admit it, toilets (hopefully) play a big part in your life. But how do they work?

Though it may seem like magic, the toilet's disappearing act is straightforward, based on elementary principles of fluid mechanics. Here we'll develop the fluid principles we need, and apply them to understand humanity's most widely used invention.

The toilet operates via a chain of effects in fluid mechanics. The crucial ideas we'll need to explain it are:

  • driven fluid flows.
  • the hydrostatic pressure of standing water.
  • the siphon.

The first effect we'll study is how to drive fluid flows in the first place. Consider the pressure on the left- and right-hand side of the fluid below. What would cause the water to flow left to right across the pipe?

Now we study the hydrostatic pressure. Suppose you are swimming \(\SI{1}{\meter}\) underwater and then dive to \(\SI{20}{\meter}.\) Will you feel a change in pressure?

The siphon is an old technology used to transport water from higher to lower locations, and is commonly used to clean fish tanks, transport water for irrigation, and to steal gasoline. If a tube is completely filled with water (i.e. it has no air gaps), then hydrostatic pressure (due to gravity) at the high end will drive water to the low end. This is known as the siphon effect.

Importantly, it does not matter if the tube goes up before it goes down to the lower end, and all that matters for the siphon effect is the relative height of the ends.

We've now established three important principles:

  • Fluid pressure increases with depth in any column of water.
  • Fluid flows when it is subject to a difference in pressure.
  • A siphon can transport water from high to low points.

Simple as they are, these ideas are at the core of toilet functionality. Now that we have the theory laid out, let's build up the mechanics of the flush.

Suppose we slowly add water to the toilet bowl below—e.g. by pouring it in from a bucket. What will happen to the water level in the bowl?

Suppose we fill the bowl quickly so that water moves up and over the U-tube, forming a seal as shown below, what would happen next?

We now have everything we need to understand the function of the common household toilet.

  • When we flush, the tank on the back releases water that flows out under the pull of gravity.
  • This fills the bowl quickly (in about \(\SI{5}{\second}\)), raising the water pressure in the bowl.
  • This creates a large difference in pressure between the water in the U-tube and the water in the bowl, driving water up and over the U-tube, into the sewer pipe.
  • As we saw, this forms an airtight seal between the water and the tube, which turns the bowl:U-tube into a siphon.
  • The siphoning then drains the entire contents of the bowl into the sewer pipe, after which the siphon seal breaks and the bowl is empty.

Now, we'll wrap up with some consideration of design choices in the toilet.

Up to now, the toilet we've described is the simplest manifestation of the jet-less siphon toilet, which was popular in the United States until the late 1990's. As they require a large volume of water \(\left(\SI{3.5-5}{\liter}\right)\) to initiate a flush, they were phased out of use.

In their stead are so-called siphon jet toilets. Whereas jet-less toilets depend on the rising hydrostatic pressure of water in the bowl to create the siphon, the siphon jet toilet directs most of the water flow from the tank toward a small outlet just in front of the U-tube. This causes a very fast flow rate over the U-tube that can kick-start the siphon much faster than a jet-less toilet, and with much less water.

Does the height of the water tank have a significant effect on the pressure that initiates the flush in a jet-less toilet?

Suppose you could build a toilet that is the same as your normal toilet, but the bowl is 3 ft tall, so that the height of the water column that rushes in during the flush is higher as well (as shown in the diagram). What impact would this have on the formation of clogs?

In this quiz, we laid out three basic principles of fluid mechanics and applied them to understand the workings of the toilet.

  • Fluid pressure increases with the height of any fluid column.
  • Fluids flow down pressure gradients.

The second principle is a special case of what is known as Bernoulli's principle, which relates the pressure, potential, and kinetic energy of a fluid in pipes. In the general case, these mechanics are governed by the Navier-Stokes equations.

Fluid mechanics is one of the more universally applicable aspects of physics, and applies to much more than just toilets. Continuing on with the principles touched on here will lead us to understand global weather cycles, airplane turbulence, and even the course of hurricanes.

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