Has your house ever lost power?

Modern electrical infrastructure uses incredibly complex systems with many possible points of failure: faults at power generating stations, mechanical damage to electrical transmission lines, short circuits in distribution stations, or overloading of high voltage electricity main lines. It shouldn’t be surprising that winter storms, high wind, or even high demand on a hot day lead to temporary power outages.

But do you ever remember your house losing water pressure, even when the electricity has gone out? Probably not, since water distribution in most of the developed world is powered by a omnipresent force that never goes out: gravity.

The water pressure coming into a typical home is around \(\SI{400}{\kilo\pascal},\) equal to 400 thousand kilograms per square meter of pressure. This is enough to provide plenty of water pressure out of a typical shower head, and run household appliances like a dishwasher or a laundry machine. You might expect that this pressure is generated by mechanical pumps at a water distribution hub much like electricity is generated by power plants.

But mechanical pumps aren’t the only way to generate water pressure. As we learned in a previous quiz, water pressure increases with depth. When you dive under the water in a swimming pool, the pressure increases because of gravity pulling on the mass of water above you. The pressure increases by about \(\SI{10}{\kilo\pascal}\) per meter of depth. Cities take advantage of this by storing water in **water tanks** elevated far above ground level. When water flows from a water tank, gravity provides the pressure that you feel from your shower head or sink faucet.

This pressure generated from gravity pulling on the mass of water is called **hydrostatic pressure**. The pressure difference between the top and bottom of a pipe with a height difference \(\Delta h\) is given by Pascal's Law (which we'll dive deeper into soon):

\[\Delta P = \rho g \Delta h\]

where \(\rho\) is the density of water.

Consider this peculiar water tower in South Carolina, USA. It’s called the Peachoid to celebrate the region's quality peaches. The tower sits just outside of the town of Gaffney, and provides water pressure for the entire town.

If the citizens of Gaffney want standard \(\SI{400}{\kilo\pascal}\) water pressure in their homes, how tall does the water tower need to be to provide that pressure when full to a home at the foot of the tower?

**Remember** that the density of water is constant, and roughly equal to \(\rho = \SI[per-mode=symbol]{1000}{\kilogram\per\meter\cubed}.\)

To figure out how Pascal came up with this law, let’s look at a small vertical section of a pipe with area \(A\) and height \(h\) filled with water. At the moment, the water isn’t flowing, but we can still calculate the static (unmoving) pressure of the water at the top and bottom of this section.

Since the water isn’t flowing, the three forces acting on this column of water must be balanced. Since pressure \(P\) is just a force applied over an area, the top and bottom forces can be expressed using the pressure of the water at those points.

\[\begin{align} \sum F &= F_{\textrm{top}}+F_{\textrm{bottom}}+F_{\textrm{weight}} =0\\ \sum F &= -P_{\textrm{top}} A + P_{\textrm{bottom}} A + F_{\textrm{weight}}=0 \\ \frac{F_{\textrm{weight}}}{A} &= P_{\textrm{top}} - P_{\textrm{bottom}} \end{align}\]

The weight of the water column is derived from its mass and the acceleration of gravity \(F_{\textrm{weight}} = m g\), but can also be broken down using the column’s volume and density as \(F_{\textrm{weight}} = \rho A h g.\)

\[P_{\textrm{top}} - P_{\textrm{bottom}} = \rho g h\]

Putting these together we find a very unexpected result. The **pressure difference** between the top and bottom of the pipe doesn’t depend on the volume of water, only on the **difference in heights**. We’ll see how the consequences of this result are applied to water distribution throughout this quiz.

The Peachoid stands exactly 135 feet (or \(\SI{41.14}{\meter}\)) in height, and it’s situated a few kilometers outside of the center of town. A large feed pipe from the water tower leads to the city’s plumbing infrastructure, and a series of sealed main pipes carry the water all the way into town. The plumbing infrastructure acts like a large siphon, collecting water at a higher elevation from the water tower, and delivering it to homes.

Given the different paths of sealed pipes shown below, which house in Gaffney do you think would have the highest water pressure?

The tanks of municipal water towers are typically quite large. While a normal in-ground swimming pool in a back yard might hold approximately 100 000 liters of water, a typical water tower (like the Peachoid) holds over four million liters. In hilly areas, construction of a massive water tower like the Peachoid isn’t necessary: towns and cities build underground cisterns or reservoirs at higher elevation which serve the same purpose as water towers and provide gravity-fed water pressure.

Tanks and reservoirs are normally sized to hold about a day’s worth of water for the town using the water tower. If the power goes out, or the city’s water supply gets cut off, the water tower holds enough water to keep things flowing for another day. The Peachoid’s 4 million liters of water are stored in a spheroid nearly 20 meters in diameter.

The Peachoid is designed to hold enough water for a typical day in Gaffney. Its 4 million liters of water averages out to about 2800 liters per minute over the course of the day. This might seem like a lot of water, but a typical shower head uses about 8 liters per minute, meaning only about 350 people could shower at any given time in Gaffney.

This doesn’t make much sense, since over 12,000 people live in the area served by the Peachoid water tower.

All we were able to calculate from the Peachoid’s capacity was Gaffney’s average usage: What do you expect the usage of water to look like over the course of a day?

Most cities and towns don’t just use water towers to provide pressure, they also use them to accommodate peak pressure usage. Every morning when the population of Gaffney takes a shower, the water usage peaks far above the average. If the town used mechanical pumps to provide the pressure, they would have to be able to pump the 10 000 liter per minute peak usage to provide enough pressure for everyone’s morning showers.

There is a very large cost difference between a 2800 liter per minute pump that can fill the water tower once per day, and a 10 000 liter per minute pump that could keep up with peak usage. A slow and steady pump fills the Peachoid when usage is low, and when the pumps are inadequate during the morning rush, the pressure from falling water picks up the slack.

So far we’ve discussed water towers that feed towns and small cities, but what about big cities like New York? The principle is the same: New York’s reservoir is located in Delaware County, an area that’s significantly elevated with respect to Manhattan.

But it turns out that the effect of height on hydrostatic pressure is a double-edged sword. The same force of gravity that pushes water flowing from an elevated tank to a ground level house pulls water flowing from ground level up to a 10th floor apartment. If a city’s reservoir provides \(\SI{400}{\kilo\pascal}\) of water pressure at ground level, that pressure decreases by the same 10 kPa per meter of elevation.

The Delaware County reservoir provides a water pressure of \(\SI{540}{\kilo\pascal}\) to the island of Manhattan. If a building in the city relies solely on this reservoir to service its floors, at what floor will the hydrostatic pressure fall to zero?

Assume that each floor of the building has a height of \(\SI{3}{\meter}.\)

To solve this problem, most buildings above six stories in New York City are required to have their own water towers to provide pressure for the building’s upper floors. These large, cylindrical tanks are a common sight on the skyline of Manhattan and are usually found on the roof of the building (for obvious reasons).

Since the city water pressure isn’t sufficient to bring water up to the roof, each building generally needs its own mechanical pumps to bring the water up to the roof of the building. If this is the case, why don’t the buildings just pump the water to everyone’s apartment?

The physical laws which define hydrostatic pressure and the equation we’ve used to calculate it were first “discovered” by Blaise Pascal in the 17th century, and Pascal’s law still bears his name:

\[\Delta P = \rho g \Delta h\]

After deducing that a column of water generates a pressure dependent only on its height and not on the volume or mass of water, the story goes that Pascal decided to perform a unique experiment that is now referred to as Pascal’s paradox. He filled a barrel with water and attached a very long narrow tube to its mouth, which he filled with water. Though the volume of water in the tube was only a small increase compared to the large amount of water in the barrel, the additional height created a giant pressure on the barrel, causing it to leak and then explode.

Illustration from Amédée Guillemin's "The Forces of Nature," 1872.

Despite them bearing his name, the principles of Pascal’s law and paradox did not begin with Blaise Pascal. The Romans commonly used aqueducts to provide indoor plumbing to the second or third floor in Rome from elevated aquifers many miles away over 1500 years before Pascal. Romans were even known to use the same barrel-breaking apparatus while mining. They would fill deep narrow cavities in mountains and generate immense pressure to shatter thick stone walls.

The simple principles of hydrostatic pressure that served the Romans so well continue to see use for gravity-fed water distribution in many developed and developing countries, including the entire cities of New York and San Francisco.

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