What If Everyone in Japan Flushed at the Same Time? We Ran the Numbers | What If, Calculated #05

The answer in 30 seconds

The first thing to reach its limit is not the biggest facility. It is the small street pipe near your home.

On a model city of 100,000 people, if every home flushes within 10 seconds, flow in those small pipes reaches 4.4 times their limit, and they stay completely full for about 39 seconds. Flush at exactly the same instant and it is 12 times the limit. The treatment plant, meanwhile, peaks at only 59% of the amount it is built to expect — everyday wastewater included. The wave weakens as it moves into larger pipes.

In this model, spread the flushes over about a minute and a half and the small pipes stay within their limit (0.87×). On the supply side, each tank's slow 75-second refill acts as a cushion: the reservoir drops by only 1.8%.

If you have ever heard of the "Super Bowl flush" — the halftime surge in water use in the United States — this episode is that idea taken to its limit, with the underground math done properly.

Assumptions (conditions and data)

The calculation has two layers, and we keep them separate.

• A. Model city: a city of 100,000 people where we compute flow/limit for each level of pipes. The "what fails first" answer comes from here.
• B. National estimate: only the total volume for all of Japan. We do not model every pipe in the country — they differ by region.

Conditions we assumed

The pipe hierarchy

Sewers collect like a river system: small street pipes (200 households each) join medium pipes (2,000), then large mains (10,000), then the treatment plant. Supply works in reverse: reservoir, large mains (10,000), medium pipes (2,000), small supply pipes (200).

The formulas

Sewer limits (Manning's equation)

The full-pipe flow of a circular pipe is given by Manning's equation:

Q = (1/n) × A × R^(2/3) × √S

• Q: flow (m³/s)
• n: roughness coefficient (0.013)
• A: cross-sectional area (πD²/4, where D is diameter)
• R: hydraulic radius (D/4 when full)
• S: slope

That gives limits of 32.6 L/s for the small street pipe (250 mm, slope 3‰), 237.8 L/s for the medium pipe (600 mm, 1.5‰), and 758.2 L/s for the large main (1,000 mm, 1‰).

Routing the wave (storage routing)

One household's flush is a triangular wave: 6 liters over 6 seconds. We add the everyday wastewater that is always flowing (the daily average). Each pipe passes water at most at its limit; anything above that is stored in the pipe (it runs full) and drains later. Flows from child pipes are summed with travel-time delays (distance ÷ 1 m/s). Because pipe lengths differ, arrivals spread out and the wave gets lower and wider. That spreading is why the wave weakens on its way to the plant.

The supply side

Tank refills are modeled as a flat 75-second draw; the direct-from-main hypothetical as a flat 20-second draw. We compare the summed toilet-only demand with the usual hourly peak and with a 2 m/s velocity guideline. We do not compute water pressure itself. The reservoir is represented by 12 hours of the planned daily maximum (15 million liters).

The planning-stage program was re-run at production time and every value matched. The web version reimplements the same formulas in JavaScript, verified to match the reference ratios to two decimal places for Experiments 1–3.

How to read the results

Three points matter.

1. Smaller pipes have less headroom, so they hit their limit first. One small pipe serves 200 households, and a synchronized wave is simply too big for it.
2. Large pipes and the plant are fine. Arrival-time spreading flattens the wave; the plant's peak arrives about 37 minutes late at 59% of expected. Note that everyday flow alone accounts for about 33 points of that 59%; the flush wave adds about 26 points.
3. On the supply side, demand spikes but the water does not run out. Refill demand reaches 3.5× the usual busy hour, yet the reservoir drops only 1.8%. The slow tank refill protects the city.

The dividing line for the small pipes (all homes flushing): 12× at 0 s of spread, 4.4× at 10 s, 2.0× at 30 s, 1.13× at 60 s (still over), 0.87× at 90 s, 0.58× at 180 s.

Nationally, 55.7 million households × 6 L is about 330,000 m³ — roughly 835 pools of 25 m (at 400 m³ each). Compared with an assumed national supply of 300 L per person per day, that is about 0.9%. The volume is small; the speed is the problem.

As references: with older 13 L toilets flushed within 10 seconds, the small pipes reach 9.5× (full for about 83 s) and the plant 74% — small pipes still first. Changing the flush duration to 4 s or 8 s moves Experiment 2's small-pipe ratio to 5.25× or 3.92×; the ordering does not change. A separate building-only calculation for a 100-home apartment drain stack (100 mm, about 4 L/s guideline) gives a simultaneous peak of 50× the guideline.

Where the model differs from reality

• The model city is not a real city. Pipe sizes, slopes, and household counts are representative values. Real numbers will differ, but the ordering — small pipes first — should hold wherever small pipes have less headroom than mains.
• Rain is not included. In combined-sewer areas on a rainy day, margins would be smaller.
• For a full small pipe, the calculation only tells us how long it stays full. We therefore say "drains poorly, with a risk of push-back at low drains," and do not claim water jets out of manholes.
• We did not compute the size of any pressure drop on the supply side, so we make no claim about weaker flushing or slower refills. What we can say is the demand multiple and that stored water barely moves. Pressure and flushing effects are simply not answered by this calculation.
• Everyday wastewater is held at its daily average. An evening peak would raise the plant percentage somewhat.
• Pump stations, treatment-plant controls, and buildings with their own receiving tanks are not modeled. Receiving tanks would cushion the supply side further.

Frequently asked questions

Would water really shoot out of manholes?

Our calculation can only say how long the small pipes stay full — about 39 seconds in Experiment 2. During that time, nearby homes would drain poorly, and low drains such as a basement bath or sink could see water push back. Whether a manhole overflows depends on local terrain and connections, so we do not claim it.

What if it were raining?

In areas with combined sewers, where rain and wastewater share pipes, a rainy day already eats into the margin, so the same flush wave would fill pipes more easily. Rain is not in this calculation, so we give no number. In separated-sewer areas, rain uses different pipes and this effect is small.

Is my city the same?

This is a model city built from representative values. Your city's pipe sizes, slopes, and households per pipe differ, so the multiples will differ. The ordering, though — the small pipe near your home has the least headroom and reaches its limit first — is likely to hold in most cities.

Does anything like this actually happen?

Yes, in milder form. Water use surges during breaks in major sports broadcasts are actually observed — New York City's Department of Environmental Protection has published about bracing for the Super Bowl halftime "super flush." The conditions differ from our model, so we cite the observation only, without comparing numbers.

Try it on the web

On this episode's web page you can change the city size, toilet type, spread of flush times, and share of homes flushing, and run the same calculation yourself. The main game: find the spread that keeps even a full-city flush within the limit.