Fixture units and drainage loads

Maya's next assignment was a tally. Luis gave her the plan of a typical floor on Kingsway Tower, eight suites around a central corridor, and asked her to count every fixture that drains to soil stack S-3 and add up its load. She started by adding gallons per minute, the way she had seen water pipes sized, and Luis stopped her. Drains, he said, are not sized by adding flows. They are sized by fixture units, and fixture units exist because no building ever uses all its fixtures at once.

This lesson explains what fixture units are, where the numbers come from, how to tally the load on a drain, how to handle fixtures that discharge continuously, and why the code's tables can size a pipe serving a hundred fixtures only a little larger than one serving ten. Lesson 4 uses these tallies to size branches and stacks, and lesson 5 to size building drains and sewers.

The problem with adding flows

A water closet can discharge at about 25 gpm for a few seconds when flushed. A bathtub draining empties at perhaps 10 gpm for a few minutes. If you added the peak flow of every fixture on Kingsway Tower and sized the building drain for that total, you would get a pipe several times larger than needed, because the chance of every toilet, tub, sink and washer in a 12-story building discharging at the same instant is effectively zero.

What matters is the flow the drain will realistically see at the busiest moment, such as weekday mornings when many people shower, flush and run taps within the same half hour. Even then, each fixture discharges only for a short time, and the discharges are scattered. The more fixtures a drain serves, the smaller the fraction that will be discharging at once.

Engineers in the early twentieth century studied this problem using probability. The best-known work is by Roy Hunter at the US National Bureau of Standards, who treated each fixture as discharging for a certain time at a certain interval and calculated how many would overlap at a peak. The fixture unit method that every North American plumbing code uses for drainage descends from that work.

What a fixture unit is

A fixture unit is a number that represents how much load a fixture puts on the drainage system compared with other fixtures, taking into account how much water it discharges, how fast and how often. It is a relative measure, not a flow rate. A lavatory, which discharges a small amount slowly and briefly, is the reference with a value of about one fixture unit. A water closet, which discharges a large slug of water and solids, has a value of several fixture units.

The Ohio Plumbing Code, following the IPC, calls them drainage fixture units, abbreviated dfu, and lists a value for each type of fixture in a table in the sanitary drainage chapter. The tables that size pipes are then written in total dfu.

Because fixture units account for probability, you can add them. If a branch serves a water closet, a lavatory and a bathtub, you add their fixture unit values and use the total to size the branch. The sizing tables then turn that total into a pipe size, with the probability of overlap already built in. That is the key idea: add fixture units, never add flows.

The practice values in the table are deliberately round numbers so you can focus on the method. Real code tables give more categories, such as different values for private and public water closets, flushometer valves, commercial sinks, and fixtures connected to a pumped system. Public fixtures get higher values because they are used more often.

Private and public use

Codes distinguish between private use, such as fixtures in a suite or a house, and public use, such as fixtures in an office washroom, a school or a restaurant. Public fixtures are used much more frequently, so their discharges overlap more and they get higher fixture unit values. A water closet in a suite on Kingsway Tower counts as private use; the same model of water closet in the ground floor amenity washroom counts as public use and carries a higher load.

Many codes also give a combined value for a group of fixtures in one bathroom, sometimes called a bathroom group: a water closet, lavatory and bathtub or shower used by the same person. Because one person rarely uses all three at the same moment, the group value is lower than the sum of the three separate values. Where a code gives a group value, use it for suite bathrooms instead of adding the fixtures one by one, but read the definition carefully, because the group must match the code's description exactly.

Tallying the load on a drain

A tally works from the top of the system to the bottom, or from the farthest fixture to the sewer, adding loads as each drain joins. For each pipe you record the fixtures it receives and the running total of fixture units. The habit of writing the total on each pipe of a sketch or riser diagram makes sizing and checking much easier.

Maya did the tally for soil stack S-3. Each suite on that stack has one bathroom with a water closet, a lavatory and a bathtub, and the stack also receives the suite's clothes washer standpipe. Using the course practice values, each suite adds 4 + 1 + 2 + 3 = 10 fixture units to the stack. Kingsway has twelve residential floors with one suite per floor on stack S-3, so the stack carries 12 × 10 = 120 fixture units at its base.

A tally is only as good as the fixture count. Check the architectural plans as well as the plumbing plans, because fixtures are sometimes added late in design, and include fixtures that are easy to miss: floor drains, hose bibb drains, condensate from mechanical equipment and laundry standpipes.

# Tally drainage fixture units for stack S-3 (course practice values)
VALUES = {"water closet": 4, "lavatory": 1, "bathtub": 2, "kitchen sink": 2,
          "dishwasher": 2, "clothes washer": 3, "floor drain": 2}

suite = ["water closet", "lavatory", "bathtub", "clothes washer"]
per_floor = sum(VALUES[f] for f in suite)
running = 0
for floor in range(12, 0, -1):
    running += per_floor
    if floor in (12, 9, 6, 3, 1):
        print(f"floor {floor:2d}: adds {per_floor} dfu, stack total below this floor {running}")
print("load at the base of stack S-3:", running, "dfu")

floor 12: adds 10 dfu, stack total below this floor 10
floor  9: adds 10 dfu, stack total below this floor 40
floor  6: adds 10 dfu, stack total below this floor 70
floor  3: adds 10 dfu, stack total below this floor 100
floor  1: adds 10 dfu, stack total below this floor 120
load at the base of stack S-3: 120 dfu

Why the pipe does not grow in step with the load

If you look at any code's drainage sizing table, you will see that the capacity of a pipe in fixture units grows much faster than its size. A pipe twice the diameter can carry far more than twice the fixture units. Two effects combine. First, a larger pipe has much more cross-sectional area and runs faster for the same slope, so its flow capacity rises steeply with diameter. Second, the more fixtures a pipe serves, the smaller the share discharging at once, so the realistic peak flow grows more slowly than the fixture unit total.

This is why the building drain at the bottom of Kingsway Tower, carrying several hundred fixture units from all the stacks, is only one or two sizes larger than the stacks themselves. It also explains why a single water closet needs a pipe as large as it does: with only one fixture there is no diversity at all, and the drain must handle the full flush.

The program below shows the idea using a simple probability model. Each fixture is assumed to discharge for a short time at random during a busy period, and the program estimates how many fixtures are likely to be discharging at once. Real code tables come from more detailed studies, but the pattern is the same: the busy-period peak grows much more slowly than the number of fixtures.

# How many fixtures discharge at once? A simple binomial model of diversity
from math import comb

def likely_peak(n, p, confidence=0.99):
    """Smallest k such that P(at most k fixtures discharging) >= confidence."""
    total = 0.0
    for k in range(n + 1):
        total += comb(n, k) * p ** k * (1 - p) ** (n - k)
        if total >= confidence:
            return k
    return n

p = 0.03     # each fixture discharging 3 percent of the time in the busy period
for n in (1, 5, 10, 25, 50, 100, 200, 400):
    k = likely_peak(n, p)
    print(f"{n:4d} fixtures: plan for {k:2d} at once ({100 * k / n:5.1f} percent)")

   1 fixtures: plan for  1 at once (100.0 percent)
   5 fixtures: plan for  1 at once ( 20.0 percent)
  10 fixtures: plan for  2 at once ( 20.0 percent)
  25 fixtures: plan for  3 at once ( 12.0 percent)
  50 fixtures: plan for  5 at once ( 10.0 percent)
 100 fixtures: plan for  8 at once (  8.0 percent)
 200 fixtures: plan for 12 at once (  6.0 percent)
 400 fixtures: plan for 21 at once (  5.2 percent)

Continuous and semi-continuous flows

Fixture units work for fixtures that discharge intermittently. Some equipment discharges continuously or for long periods: sump pumps, sewage ejectors, cooling equipment, water softener backwash and some commercial processes. A continuous flow overlaps with everything else, so the probability approach does not apply to it.

Codes handle this by converting a continuous flow into an equivalent number of drainage fixture units. The IPC and the Ohio Plumbing Code assign 2 dfu for each gallon per minute of continuous or semi-continuous flow, so a pump discharging 20 gpm adds 40 dfu to the drain it connects to.

On Kingsway Tower, the sump in the lower parking level discharges through a pump rated at about 30 gpm when the level rises. Because it runs for long periods during wet weather, the engineer counted it as a continuous flow when sizing the building drain it connects to.

Loads from storm water and from other buildings

Storm water is not counted in fixture units. Roof drains, area drains and foundation drains are sized by drained area and rainfall rate, which lesson 11 covers. Where a code allows a combined building drain or sewer, the storm load is converted to an equivalent for sizing using a method set out in the code. On Kingsway the systems are separate, so the sanitary tally contains only fixtures and pumped flows.

Loads can also come from future work. The engineer may size a building drain for a planned future tower or a future restaurant tenant on the ground floor, and that capacity shows up as a larger pipe than the present tally needs. As always, install what the drawings show.

Where tallies go wrong

Most tally errors come from three sources. First, using the wrong value: a public water closet counted as private, or a flushometer counted as a flush tank. Second, missing fixtures: floor drains, hub drains and equipment drains. Third, adding a group value and its component fixtures twice. Inspectors check tallies against the drawings, and a sizing error found at rough-in can mean replacing pipe that is already hung.