Plumbing Formula Reference
Every formula used in plumbing design and field work — water supply, drainage and venting, pressure and pump head, water heating and gas load. Each entry gives the formula, what each variable means, and the typical values you should expect back. IPC citations throughout so you can check any of it against the book.
Water Supply Fixture Units → Peak Demand (GPM)
Variables
- WSFU
- — Water supply fixture unit — a weighting per fixture type, from IPC Table E103.3(2)
- Hunter's curve
- — The probability curve in Table E103.3(3) that converts fixture units to a design flow
- hot / cold split
- — For a fixture with both supplies, each side is 3/4 of the fixture's total
Normal range / typical values
A WSFU is a probability weighting, not a flow rate — 100 WSFU is nowhere near 100 GPM, because no real building runs every fixture at once. A typical single-family house lands around 20–30 WSFU and 12–18 GPM of design demand
Pipe Velocity
Variables
- V
- — Velocity in feet per second
- Q
- — Flow in gallons per minute through that section
- d
- — Inside diameter in inches — not nominal size; they differ by material
- 0.408
- — Unit constant converting GPM and square inches to ft/s
Normal range / typical values
The IPC sets no velocity cap. Section 604.1 asks only for "accepted engineering practice" and an approved sizing method. The familiar 8 ft/s cold and 5 ft/s hot limits are ASPE and copper-tube manufacturer guidance against erosion-corrosion — good practice, not IPC code. The UPC is stricter here, so check which code your jurisdiction adopted before calling a velocity a violation
Hazen-Williams Friction Loss
Variables
- Q
- — Flow in gallons per minute
- C
- — Roughness coefficient: copper 130–150, PEX and CPVC ~150, galvanized steel 100–120, aged cast iron ~100
- d
- — Inside diameter in inches — the 4.87 exponent is why one size up cuts loss so hard
Normal range / typical values
Valid for water between 40 °F and 75 °F in turbulent flow only. The d^4.87 term dominates: going from 1/2 in to 3/4 in copper at the same flow drops friction loss roughly six-fold, which is usually cheaper than upsizing the meter
Available Pressure for Friction
Variables
- P_static
- — Street or well pressure at the meter, measured not assumed
- P_fixture_min
- — Minimum flow pressure for the governing fixture, from IPC Table 604.3
- 0.433 × height
- — Elevation penalty in psi per foot of rise to that fixture
- device losses
- — Softener, filter, backflow preventer, PRV — each takes its cut
Normal range / typical values
IPC Table 604.3 sets the floor: most fixtures need 8 psi flow pressure, a flush tank 20 psi, a flushometer valve 15–20 psi at 25 GPM. Whatever is left after all four deductions is the entire budget for pipe friction
Allowable Friction Loss per 100 ft
Variables
- P_avail
- — Pressure left over for friction, from the formula above
- L_developed
- — Total developed length to the governing fixture, fittings included
Normal range / typical values
This is the number you carry into the sizing table — pick the smallest pipe whose loss at design flow stays under it. Residential jobs usually land between 2 and 8 psi per 100 ft
Developed Length & Fitting Equivalents
Variables
- L_measured
- — Actual pipe run to the governing fixture, following the route
- equivalent length
- — Each fitting expressed as the feet of straight pipe that loses the same pressure
Normal range / typical values
Multiplying the measured length by 1.5 is the common residential shortcut. On a fitting-heavy run — a tight mechanical room, several changes of direction — the fittings can outweigh the pipe, so count them
Drainage Fixture Units (DFU)
Variables
- DFU
- — Drainage fixture unit per fixture type, from IPC Table 709.1
- Table 710.1(1)
- — Maximum fixture units on horizontal branches and stacks, 2-1/2 in and smaller
- Table 710.1(2)
- — The same limits for 3 in through 6 in pipe
Normal range / typical values
DFU and WSFU are different scales for different systems — do not mix them. The stack column and the horizontal-branch column differ for the same pipe size, because a vertical stack carries more than a horizontal run of the same diameter
Minimum Drain Slope
Variables
- 1/4 in/ft
- — The residential default — everything 2-1/2 in and under
- 1/8 in/ft
- — Permitted from 3 in to 6 in, where scouring velocity still holds
- 1/16 in/ft
- — Permitted for 8 in and larger — a size range that almost never appears in a house
- exception
- — Piping upstream of a grease interceptor stays at 1/4 in per ft
Normal range / typical values
Only the first two rows appear in residential work, since 8 in drains are a commercial and municipal size. The UPC generally holds 1/4 in per ft with allowances for larger pipe, so confirm which code your AHJ enforces before flattening a long run
Slope as Percent & Total Fall
Variables
- fall
- — Total vertical drop across the run — what has to fit in the joist bay
- run
- — Horizontal distance in feet
Normal range / typical values
1/4 in per ft = 2.08%, 1/8 in per ft = 1.04%. A 40 ft run at 1/4 in per ft drops 10 in — the number that decides whether the drain clears the footing, and the usual reason a long run goes to 3 in pipe at 1/8 in per ft
Manning's Equation — Drain Capacity
Variables
- V
- — Velocity in feet per second
- n
- — Roughness: 0.009–0.015 for plastic, 0.012–0.015 for cast iron
- R
- — Hydraulic radius — flow area divided by wetted perimeter, in feet
- S
- — Slope as a decimal, so 1/4 in per ft is 0.0208
- A
- — Cross-sectional area of the flow, in square feet
Normal range / typical values
The IPC capacity tables assume drains 3 in and larger run half full. That is why oversizing can backfire: the same flow in a bigger pipe at a flatter slope spreads thinner, velocity falls below the ~2 ft/s scouring threshold, and solids drop out instead of moving
Vent Size
Variables
- drain dia
- — The drain size the DFU table demanded — not the pipe someone installed
- min 1-1/4 in
- — The floor, no matter how small the drain
- 40 ft rule
- — Applies to the whole developed length of that vent, not just the excess
Normal range / typical values
4 in drain → 2 in vent. 3 in drain → 1-1/2 in vent. Anything 2 in and under lands on the 1-1/4 in minimum. Individual, branch, circuit and relief vents all follow the same half-diameter rule
Trap Arm Maximum Length
Variables
- max len
- — Developed length from the trap weir to the inner edge of the vent fitting
- minimum distance
- — Also a floor: at least 2 × the trap arm diameter from weir to vent
- max slope
- — Counts toward the limit — a steeper arm siphons the trap sooner
Normal range / typical values
Self-siphoning fixtures — water closets above all — are not length-limited, because the fixture reseals its own trap. Everything else is, and exceeding the table is the most common reason a trap gurgles and then dries out
Grease Interceptor Sizing
Variables
- L × W × D
- — Interior of one sink compartment in inches; 231 in³ is one US gallon
- fill
- — Fraction of the compartment assumed to drain — conventionally 75%
- drain min
- — Drainage period, normally taken as one minute
- retention hr
- — 2.5 hours for a commercial kitchen with a dishwasher, 1.5 for single-service
- storage
- — 1 for an 8-hour day, 2 for 16, 3 for 24 — a straight multiplier on the vessel
Normal range / typical values
Two devices, two units. A hydromechanical interceptor is rated in GPM and separates by slowing flow; a gravity interceptor is rated in gallons and separates by holding wastewater still. A standard 24 × 24 × 12 three-compartment sink already needs 75 GPM, near the top of the hydromechanical range — which is why full-service kitchens usually end up outside. The local FOG programme sets the minimum size and normally overrides the arithmetic below about 65 seats
Rainwater Yield & Cistern Storage
Variables
- area
- — Horizontal projection of the catchment, not slope length — rain falls vertically
- Cr
- — Runoff coefficient: about 0.95 standing-seam metal, 0.85 asphalt shingle, 0.30 green roof
- Ce
- — Collection efficiency after screen, filter and first-flush losses — commonly 0.85
- dry-spell days
- — How long the system has to run with no rain; the single biggest lever on tank size
- 24 hours
- — Untreated grey water goes septic past a day, so it cannot be stored between storms
Normal range / typical values
Two sources, two sizing rules. A cistern separates supply from demand in time, so it is sized on the gap between storms — but never past what the catchment can refill, or it just sits part-empty. Grey water cannot do that at all: it is a surge vessel that has to empty daily, so the design work moves to the distribution field. These are planning values rather than code, because nonpotable reuse falls to IPC Chapter 13 where adopted and to state and county rules everywhere else
Storm & Roof Runoff Flow
Variables
- area
- — Horizontal projection of the roof in square feet, not the sloped surface
- in/hr
- — Design rainfall intensity for the location, from the local 100-year hourly rate
- 7.48052 ÷ 12 ÷ 60
- — Gallons per cubic foot, spread one inch deep, per minute — the derived 96.25 divisor inverted
- A, R, S
- — Full-bore area, hydraulic radius and slope, as in Manning's equation above
Normal range / typical values
Runoff is strictly linear in area and rainfall — about 41.6 gpm per 1,000 square feet at 4 inches per hour — which makes this the one place in plumbing where a per-square-foot rule of thumb is safe. A 2,000 square foot roof in a 4 inch per hour storm sheds roughly 83 gpm, more than twenty times a household peak. IPC Tables 1106.2 and 1106.3 govern leader and horizontal storm drain sizing and are not reproduced here; the capacity line is a full-bore Manning check, not a code lookup
Septic Tank & Leach Field
Variables
- gal/bed
- — Design flow per bedroom per day — commonly 150, but set locally
- 2 days
- — Retention time; the tank has to hold two days of flow
- minimum
- — The published floor by bedroom count, commonly 1,000 gal to three bedrooms
- application rate
- — Gallons per square foot per day the soil will accept, from a perc test
Normal range / typical values
The minimum governs up to about four bedrooms and retention only takes over at five, so sizing on retention alone undersizes almost every house. Soil is the real variable: the same three-bedroom house needs about 375 square feet of field in sand and 1,875 in slow clay, a five-fold swing, while the tank does not change at all. None of this is IPC — septic falls to the IPSDC where adopted and to state and county health departments everywhere else, and it varies more than anything else in plumbing
Grey Water Yield & Irrigable Area
Variables
- shower / lav
- — Gallons per person per day from each fixture — about 25 and 5
- washer
- — Gallons per load, divided over the week to reach a daily figure
- 24 hr
- — The hold limit: untreated grey water goes septic past a day
- demand rate
- — Irrigation demand in gallons per square foot per week for the planting
Normal range / typical values
Maximum storage always equals exactly one day of yield, which is the whole difference from a rainwater cistern — grey water is a surge vessel that must empty daily rather than a store that bridges between events, so the design work moves to the distribution field. A three-person household with a washer yields roughly 104 gallons a day, about 72% of it from the shower alone. Kitchen sink and dishwasher are excluded because they are black water in most jurisdictions. Nonpotable reuse is the least code-governed subject in plumbing
Pressure ↔ Head of Water
Variables
- 0.433
- — psi gained or lost per vertical foot of water, at 60 °F
- 2.31
- — The reciprocal — feet of head per psi
Normal range / typical values
The most-reached-for conversion on this page. Both constants shift slightly with temperature, but 0.433 and 2.31 are accurate enough for any domestic system
Static Pressure Loss from Elevation
Variables
- height
- — Vertical rise from the pressure source to the fixture
Normal range / typical values
Roughly 4.3 psi to reach a second floor, 8.7 psi to a third — spent before a single foot of friction. On a 45 psi supply that is a fifth of your budget gone to gravity alone
Total Dynamic Head
Variables
- static lift
- — Vertical distance the pump actually raises the water
- friction head
- — Pipe and fitting losses at design flow, converted to feet
- pressure head
- — Pressure needed at the outlet, converted at 2.31 ft per psi
- velocity head
- — V² ÷ 2g — usually small enough to ignore in domestic work
Normal range / typical values
TDH and flow together pick the pump off its curve. Sizing on lift alone is the classic error: a well pump lifting 80 ft through a long, undersized run can easily see 160 ft of TDH
Pressure-Reducing Valve Threshold
Variables
- 80 psi
- — The static ceiling for building water distribution piping
- PRV
- — Approved valve conforming to ASSE 1003 or CSA B356, with strainer
- exceptions
- — Service lines to sill cocks and outside hydrants
Normal range / typical values
Measure static pressure at night, when mains pressure peaks — a system that reads 78 psi at 4 pm can sit well over 80 psi at 4 am. High static pressure is also what drives fixture and water-heater failures long before it trips an inspection
Thermal Expansion Volume
Variables
- V_system
- — Water heater plus piping volume, in gallons
- expansion factor
- — About 0.017 for a rise from 40 °F to 140 °F — the ratio of the two densities, minus one
Normal range / typical values
IPC 607.3 requires thermal expansion control wherever a storage water heater's cold feed passes a check valve, PRV, or backflow preventer — which is to say almost every system with a PRV. Water gains roughly 1.7% in volume over that rise, and in a closed system with nowhere to go it shows up as pressure on the T&P valve
Pressure Tank Drawdown
Variables
- P_pre
- — Pre-charge, set about 2 psi below cut-in and measured with the tank drained
- P_on / P_off
- — Cut-in and cut-out of the pressure switch — 30/50 psi is the common pair
- frac
- — Fraction of the shell delivered per cycle. Boyle’s law, so all three pressures are absolute
- run min
- — Minimum pump run per cycle — 1 minute to 10 gpm, 1.5 to 20, 2 above
Normal range / typical values
About 29.5% at a 30/50 switch, so a 20-gallon tank delivers under 6 gallons. Because it is a ratio and not a difference, the same 20 psi span gives more at a lower band — 34.5% at 20/40 against 25.8% at 40/60. Widening the band is what actually helps: 30/70 reaches 45.1%. None of this is code; it is pump-industry practice
Temperature Rise & Recovery Rate
Variables
- GPH
- — Gallons per hour the heater can raise through ΔT — its recovery rate
- 8.33
- — Pounds per gallon of water; 1 BTU raises 1 lb by 1 °F
- eff
- — Thermal efficiency: ~0.80 atmospheric gas, ~0.95 condensing, ~1.0 electric
- 3412
- — BTU per hour per kilowatt
Normal range / typical values
A 40,000 BTU/hr gas heater at 80% recovers about 43 GPH through a 90 °F rise. The same duty on a 4.5 kW electric element gives about 20 GPH — the gap that decides which one keeps up with back-to-back showers
Tankless Flow Capacity
Variables
- GPM
- — Continuous flow the unit can hold at that temperature rise
- 500
- — 8.33 lb per gallon × 60 minutes per hour
- ΔT
- — Rise from incoming ground-water temperature to delivery temperature
Normal range / typical values
Size on winter inlet, not summer. A 45 °F inlet to a 105 °F shower is a 60 °F rise, and a 199,000 BTU/hr unit at 0.95 efficiency gives about 6.3 GPM there against roughly 9.5 GPM at a 40 °F rise. Undersizing on the summer number is why tankless units disappoint in January
First-Hour Rating
Variables
- FHR
- — Gallons of hot water available in the first hour from a full, hot tank
- 0.70
- — Usable fraction before incoming cold dilutes delivery below setpoint
- GPH
- — Recovery rate, from the formula above
Normal range / typical values
FHR, not tank size, is what to compare. A 40 gal gas heater can out-deliver a 50 gal electric on first-hour rating because recovery dominates the second half of that hour
Mixing Valve / Tempered Water Ratio
Variables
- T_mix
- — Target delivered temperature
- T_hot
- — Stored temperature at the heater outlet
- T_cold
- — Incoming cold temperature
Normal range / typical values
Storing at 140 °F and delivering at 120 °F with 50 °F cold means the valve blends about 78% hot. That split is the whole point: 140 °F storage suppresses Legionella, 120 °F delivery avoids scalding, and IPC 607.1.2 governs the tempered-water control that reconciles them
Gas Load → CFH
Variables
- CFH
- — Cubic feet per hour of gas the section must carry
- BTU/hr
- — Sum of the input ratings of every appliance downstream of that section
- BTU per ft³
- — Confirm with the local utility — natural gas runs roughly 950–1,100 BTU/ft³
Normal range / typical values
Fuel gas is a separate code: sizing comes from IFGC 402.4 and NFPA 54, not the IPC. Once you have CFH, the longest-length method sets the pipe — size every section for the load it carries, read against the length of the longest run to any outlet
Pipe Volume & Gallons per Foot
Variables
- d
- — Inside diameter in inches — bore, not nominal size
- 0.0408
- — π ÷ 4 × 12 in/ft ÷ 231 in³ per gallon, all folded into one constant
- purge sec
- — Time to clear the standing volume at a given flow
Normal range / typical values
About 40 feet of 3/4 inch copper holds a gallon — the shortcut worth memorising for a purge or a chlorination charge. Because volume goes as the square of the bore, upsizing a hot line makes the wait for hot water worse: 50 feet of 3/4 inch copper is 38 seconds at 2 gpm, and the same run in 1 inch is 64
Unit Conversion
Variables
- from / to factor
- — Each unit's factor to its category's base — gpm, psi, gallons, inches, ft/s
- 1 imperial gal
- — 1.2009 US gallons — a 20% error on any spec sheet that just says gallons
- 1 bar
- — 14.5038 psi exactly, because a bar is defined as 100 kPa
Normal range / typical values
Multiply into the base unit and divide out of it, and one formula converts any pair inside a category. Most of these factors are exact definitions and carry no error: the US gallon is 231 cubic inches, the cubic foot is 7.48052 gallons, the litre and the bar are defined. The exception is psi to feet of head, which is a measurement of water density and shifts slightly with temperature — the only conversion on this page that moves
Leak & Water Waste Cost
Variables
- 15,140
- — USGS drips per gallon; reconciles with a 0.25 mL drop
- ÷ 1000
- — Utility rates are quoted per 1,000 gallons — the term most often dropped
- hot
- — Fraction of the leak that was heated, 0 to 1
- 100,000
- — BTU per therm; use 3,412.14 per kWh for an electric heater
Normal range / typical values
One drip a second is about 5.7 gallons a day and $26 a year on cold water — but $47 if it is hot on gas, because you paid to heat it before it escaped, and energy is then roughly 45% of the bill. The scale is what surprises people: a visibly running toilet at 1,000 gallons a day runs to about $4,563 a year, some 175 times the dripping tap, and gets through the EPA's whole-household 10,000 gallon figure in ten days
Job Price — Margin, not Markup
Variables
- overhead
- — Business cost carried by the job, as a fraction of direct cost
- margin
- — Applied by DIVIDING by (1 − margin) — the profit share OF THE PRICE
- markup
- — Applied by MULTIPLYING — the addition ON TOP OF cost, a different number
Normal range / typical values
A 25% markup is exactly a 20% margin, and on a $2,760 break-even that is $230 of profit that never arrives. The markup needed to hit a target margin is margin ÷ (1 − margin): 25% margin needs a 33.3% markup, 40% needs 66.7%. Pricing at the zero-overhead figure while actually carrying 15% overhead turns an intended 25% margin into about 10%
Loaded Labor Rate
Variables
- 2,080
- — Paid hours in a year — NOT the hours you can invoice
- burden
- — Payroll tax, insurance, workers' comp, benefits — commonly 30–35%
- billable hrs
- — Hours actually invoiced; 1,560 is 75% utilisation
Normal range / typical values
A $38 wage carrying 32% burden and $11,000 of overhead needs about $123 an hour at 40% margin — roughly 3.2 times the wage, which is the multiple that surprises people. Utilisation moves it further than pay does and in the more useful direction: at the same wage, 1,000 billable hours needs $164 an hour and 1,800 needs $91. Raising the wage 20% only moves the rate about 13%
Whole-House Repipe Cost
Variables
- access
- — 0.8 open walls, 1.0 drywall and fishing, 1.6 plaster, tile or slab
- $/ft²
- — Roughly 5 for PEX, 6 for CPVC, 9 for copper, before access
- region
- — 0.85 rural to 1.5 major coastal metro
Normal range / typical values
Access is a multiplier, not an addition, so the levers compound: PEX through open walls against copper through plaster is about $9,150 against $27,900 on the same 1,800 square foot house — roughly three times. Cost per square foot falls as the house grows, because permit and drywall repair are fixed, so quoting a flat rate per square foot loses money on small jobs
Water Heater Replacement Cost
Variables
- upgrades
- — Expansion tank, pan, seismic strapping, vent, gas line, condensate drain
- hours
- — About 4 for a tank swap, 6 to 8 for a heat pump or tankless conversion
- ±18%
- — A wider band than the ±15% used elsewhere — what is behind the old unit is unknown
Normal range / typical values
A like-for-like 50 gallon gas swap with an expansion tank and a pan runs about $2,070, of which the appliance is only around 53% — usually the opposite of what a customer expects. IPC 607.3 requires thermal expansion control on any closed system, so the expansion tank is rarely optional. A gas tankless conversion typically adds a vent upgrade, a gas line upsize and a condensate drain, taking a swap into project territory
Sources & standards: International Plumbing Code (IPC) 2021 — 604.1, Table 604.3, 604.8, 607.1.2, 607.3, 704.1, Table 709.1, Tables 710.1(1) and 710.1(2), 906.2, Table 909.1, 916.2, and Appendix E Tables E103.3(2) and E103.3(3). Gas piping figures are from the International Fuel Gas Code (IFGC) 402.4 and NFPA 54, which are a separate code from the IPC. Velocity limits, fitting equivalent lengths, and Hazen-Williams and Manning coefficients are ASPE and manufacturer design practice, not code requirements.
Plumbing code adoption is split: much of the country is on the IPC, while other states use the Uniform Plumbing Code (UPC), which differs on fixture-unit tables, drain slope allowances, and velocity limits. Confirm which code your jurisdiction has adopted, and which edition. Local amendments override the model code, and a licensed plumber plus the AHJ have final say on anything installed.
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