Why Drainage Pipe Is Sized for Partial Flow and Not Full

Why this matters

The empty part of a drain pipe is a component. It is the return path for the air the water displaces, and it is the volume that absorbs a pressure excursion before the excursion reaches a trap seal. Size a drain to run full and you have not built a bigger drain, you have built a piston: the water now pushes a column of air ahead of it and pulls a column behind it, and the only compliant openings in the system are the trap seals. Everything a customer calls a venting complaint is really a report about how much free air area the system had left at the moment they flushed something.

Two geometries, two different partial-flow rules

The rule is not one rule, and the two versions come from different physics. Keep them separate or you will carry a coefficient into a geometry it was never derived for.

In a vertical stack, water entering from a branch is thrown against the wall and travels down as an annular sheet with a continuous core of air inside it. The stack capacity tables in the model codes are built on a design occupancy in which the water occupies roughly a quarter to a third of the cross-sectional area, with the rest held open as that air core. That fraction is a design basis for a wetted-wall annular flow in a straight vertical pipe. It is not a measured ceiling and it does not describe a horizontal run at all.

In a horizontal branch or drain, there is no annulus. It is open-channel flow: water in the bottom of the bore with a free surface, air above it, and the design target expressed as a depth ratio rather than an area ratio, commonly at or below half the internal diameter at peak. Take the exact depth basis from the plumbing code your jurisdiction has adopted and amended, because the model documents differ on it and jurisdictions amend.

The two rules are related but they are not interchangeable, and the conversion between a depth ratio and an area ratio in a circular pipe is not linear, which is the point of the arithmetic further down.

Terminal velocity, and why a taller stack is not a faster one

Water falling in a stack does not keep accelerating. Wall friction and the drag of the air core balance gravity within roughly one to two storeys of the entry point, after which the sheet travels at a steady terminal velocity for the rest of the drop. This is why a stack's capacity is set by its bore and not by the building height, and why a twelve-storey stack and a three-storey stack of the same size carry similar loads per branch interval.

It also tells you where the trouble is. If velocity is constant down the length, the interesting events are at the ends and at the discontinuities: where water enters, where the stack offsets, and where it turns horizontal at the bottom.

The air is not a spectator

Falling water drags air with it. That entrained air has to be replaced at the top of the stack and disposed of at the bottom, and the pressure gradient that appears in the stack is the system's way of asking for that.

• Above the flow the pressure goes negative, because air is being carried away faster than the terminal is supplying it through whatever resistance the vent path has.
• Below the flow the pressure goes positive, because the air being dragged down has to leave, and at the base it meets a bore that is now partly occupied by decelerating water.

Both excursions are proportional to how fast the water is moving and how little air area is left for the air to move through. That is the whole reason free area is a design quantity: it is the flow area of the air circuit.

  open terminal (air in and out)
        |
  +-----+-----+
  |w |     | w|  annular sheet of water on the
  |a |     | a|  wall, continuous air core in
  |l |     | l|  the middle, both moving down
  |l |     | l|
  +--+-----+--+
  |###########|  hydraulic jump at the base:
  +-----------+  bore runs full, air path shut
    building drain, positive pressure below

When the free area closes

At the base of a stack the falling sheet turns horizontal and decelerates, and the flow depth jumps. That hydraulic jump can fill the bore for a short distance, and while it does, the air circuit is cut. Air arriving from above has nowhere to go and the pressure below the jump rises sharply. This is the mechanism behind the classic complaint of a first-floor fixture blowing its seal when an upper-floor water closet discharges, and it is why codes restrict what may connect to a stack within a distance of its base and require relief at that point.

The same thing happens in a horizontal line, more slowly. A branch running at full bore becomes a moving plug. Ahead of the plug the air is compressed, behind it the air is rarefied, and both ends of that go looking for the nearest trap.

Worked example: how fast the air path closes

Set the geometry first. Take a nominal 3 inch stack and use an internal diameter of 3.0 inches for the arithmetic. Schedule 40 plastic actually runs slightly over 3 inches inside, so the areas below are understated by about 3 percent, which is a correction in the safe direction and is stated here so it does not get discovered later as an error. Full bore area: pi x 1.5 squared, which is 7.07 square inches.

At the stack's design occupancy. Take the water at 29 percent of the area, a figure that belongs to a vertical annular flow and to nothing else. Water area: 0.29 x 7.07 = 2.05 square inches. Air core area: 7.07 minus 2.05 = 5.02 square inches, or 71 percent. Expressed as an equivalent bore, the air core is the square root of 0.71 times 3.0 inches, which is about 2.53 inches. So a 3 inch stack at design load contains what amounts to a 2.5 inch air duct running the height of the building.

Now let one branch discharge heavily and put the water at 60 percent of the area. Water area: 0.60 x 7.07 = 4.24 square inches. Air core: 2.83 square inches, an equivalent bore of the square root of 0.40 times 3.0, about 1.90 inches. The water occupancy roughly doubled and the air path lost 44 percent of its area, dropping from 5.02 to 2.83 square inches. Air moving through a smaller passage at a higher velocity develops a much steeper pressure drop, so the pressure excursion the traps see is not doubled, it is worse than doubled. Nothing has blocked. Nothing is broken. The system is simply being asked to move more air through less of it.

Switch geometry and print the switch. The 29 percent figure above cannot be carried to a horizontal branch, because that number was derived for an annular sheet on a vertical wall. Horizontal runs are described by depth ratio, so the arithmetic changes basis here. Take a 4 inch horizontal branch, internal diameter 4.0 inches, full bore area pi x 2 squared, which is 12.57 square inches.

At half depth the pipe is half full by area as well, which is the one depth where the two ratios coincide: water 6.28 square inches, air 6.28 square inches.

Raise the depth to three quarters of the diameter, a 50 percent increase in depth. In a circular section, three-quarter depth corresponds to about 80 percent of the full area, not 75. Water area: 0.80 x 12.57 = 10.06 square inches. Air area: 12.57 minus 10.06 = 2.51 square inches, or about 20 percent. So a 50 percent increase in depth cost the air path 60 percent of its area, falling from 6.28 to 2.51 square inches. The geometry of a circle is what makes this so unforgiving: the last quarter of the depth eats most of what is left.

What this predicts, and what a person who skipped it would say. Someone reasoning linearly assumes a branch at three-quarter depth is running at 75 percent of capacity with 25 percent in hand. In area terms it is at 80 percent, and in air terms it has 20 percent of its free path left. That is why a branch that has been "fine for years" fails the first time two fixtures coincide: the failure is not in the water, it is in the collapse of the air passage that the water's own depth causes.

The failure mode, stated concretely. A branch sized so that peak flow reaches the crown seals the vent connection along that run. The fixtures upstream of the advancing plug lose their air supply and siphon; the fixtures downstream of it get positive pressure and burp. Same event, two opposite symptoms in two rooms, which is exactly the report that gets logged as "the plumbing is haunted" and dispatched as two separate calls.

What changes the answer

The design occupancy fraction is not a physical constant and two conditions genuinely move it. A stack with an offset loses the clean annular sheet at the offset and behaves for a distance below it like a fresh entry point, which is why offsets get their own relief requirements in adopted codes. A stack fed by a single very large discharge rather than by many small ones sees its peak occupancy set by that one event, so the diversity that the fixture unit tables assume does not protect it.

How to verify you got this right

• Ask which geometry any occupancy number came from before using it. Area fraction belongs to a stack. Depth ratio belongs to a horizontal run. They agree only at half depth and nowhere else.
• Check the depth-to-area conversion rather than assuming it is linear. Three-quarter depth is about 80 percent of area, and that gap is where the free air path disappears.
• When a system reports opposite symptoms in two locations at once, treat it as one event with two ends rather than two faults. Positive at one fixture and negative at another is the signature of a bore that closed somewhere between them.
• Confirm the internal diameter you used and state where it came from. Nominal size and internal diameter are not the same number, and an area error squares whatever the diameter error was.

References

• The plumbing code as adopted and amended by the local jurisdiction, for stack and horizontal branch capacity tables, stack offset relief and the restricted zone at the base of a stack; the model documents are the IPC (ICC) and the UPC (IAPMO), neither of which binds on its own
• Trade-standard open-channel hydraulics practice for circular sections, for the depth-to-area relationship used above
• See related: What Slope Buys You and What Too Much of It Costs; How to Size a Branch From Fixture Units Without Guessing