Plumbing Pipe Sizing: Method

Plumbing Pipe Sizing

The tool sizes the water-supply main, the drainage stack with its vent and the building drain of a building from the physics the plumbing codes' tables descend from, so that every number can be traced to an equation and a public document. The demand side is Roy B. Hunter's probability method of 1940 (NBS BMS 65), the pressure budget is his 1941 procedure (BMS 79), the stack and vent follow Wyly and Eaton's 1961 analysis (NBS Monograph 31), and the horizontal drain is Manning's formula. The code tables of fixture units, pipe sizes and vent lengths are not reproduced; the fixture weights, probabilities and flows are entered, with Hunter's own values as defaults, and the analyst may substitute the values of the code in force.

Hunter's probability method

A fixture that is used once every T seconds and draws water for t seconds of each use is found flowing at a random instant with probability p = t/T. Among n such fixtures the number r in use at the same instant follows the binomial law, P(r) = C(n, r) p^r (1 − p)^n−r (BMS 65 eq. 2). Hunter defined the design factor m as the number of fixtures that is exceeded only a small fraction τ of the time of congested use, Σ_{r > m} P(r) ≤ τ, and chose τ = 1 percent: the system then gives wholly satisfactory service 99 percent of the time and the overload, when it comes, is most likely one fixture. The design demand of n like fixtures is m q, with q the average flow to one fixture while it operates.

Hunter's values (BMS 65 Table 1) are a flush-valve closet drawing 27 gpm for 9 s once in 5 minutes (p = 0.03), a flush-tank closet refilling at 4 gpm for 60 s in 5 minutes (p = 0.2), and a bathtub or shower drawing 8 gpm with p = 1/15 (60 s in 15 minutes or 120 s in 30). He tabulated, for each design factor, the smallest number of fixtures at which the probability of exactly that many in use first reaches 1 percent, and drew his curves through those points (his Table 2 and figures 2 to 5). The tool computes the same construction exactly for any p entered: the binomial tail by direct summation, the points of Hunter's Table 2, and the straight-line interpolation between them that his figures show. For a single kind of fixture it also prints the exact integer m at the nearest whole n, which is the step the curve smooths, and the normal approximation m ≈ np + z √(np(1 − p)) with z = 2.326 at 1 percent, the form Wistort proposed in 1994 and the NIST / IAPMO Water Demand Calculator uses for large buildings.

The computed curve reproduces Hunter's own readings of his curves (BMS 65 Table 9) within 2 percent from about 12 fixtures upward and his worked examples (1,325 fixture units giving 247 gpm). Below about 10 fixtures Hunter drew his curves above the probability points to merge them smoothly (his readings at 2, 4 and 8 flush valves are 7 to 10 percent above the computation, and the flush-tank branch of his design figure 5 lies about 7 percent above the probability curve near 600 fixture units). The "Hunter's drawn curve" option therefore reproduces his Table 9 readings by log-log interpolation, which is what the code tables were built from, while the computed option is the first-principles value; the chart shows both.

Mixed fixtures and fixture units

Fixtures of different kinds operating independently have a joint distribution that Hunter judged too laborious for the office, so he weighted each kind against the flush valve. The weights are inversely as the numbers of fixtures of each kind producing the same demand (BMS 65 Table 3 at 150 to 300 gpm): 10 for a flush-valve closet, 5 for a flush-tank closet, 4 for a bathtub or shower, 2 for a lavatory from a consideration of its outlet, with private (one person at a time) service about half of public service (BMS 65 Table 7, BMS 79 Table 6). The total of n × weight is referred to the flush-valve curve (10 units per closet) or, where flush tanks predominate, to the flush-tank curve (5 units per closet); the tool prints each row's derived weight at 200 gpm from the same construction so that an entered p and q can be checked against the scale.

The tool also computes the exact joint distribution of all rows that carry p and q by convolving their binomial distributions on a flow grid, which is the exhaustive computation Hunter described and the convolution option of the Water Demand Calculator. For Hunter's own fixtures the two agree closely: fifty flush valves with fifty bathtubs give 167 gpm exactly and 167.7 gpm by the fixture-unit method. Rows without p and q (lavatories, sinks) enter only the fixture-unit sum. Continuous demands, hose bibbs, irrigation and cooling make-up, are added to the fixture demand as Hunter directs. The NIST GCR 22-036 records that measured modern residential use gives much lower parameters than Hunter's congested service, a tank toilet at p = 1 percent and 3 gpm rather than 20 percent and 4 gpm; such values may be entered per row, and the exact joint value then shows their effect.

Pressure budget and supply pipe

Following BMS 79, the pressure the main delivers is spent on the static lift to the highest fixture (ρ g h, 0.434 psi per foot), the loss through the water meter, the pressure the highest fixture needs to work (15 psi for a flush valve, 8 to 10 psi for faucets and flush tanks), and friction in the pipe. The remainder divided by the developed length plus the equivalent length of the fittings is the friction rate the pipe may run at. The disk-meter loss is taken as 25 psi at the rated flow of the 1941 AWWA specification Hunter quoted (BMS 79 Table 8) varying with the square of the flow, or entered. Friction is computed by Hazen-Williams, h = 10.67 L Q^1.852 / (C^1.852 D^4.87) in SI units with C = 150 for copper and 100 for steel in service, or by Darcy-Weisbach with the Colebrook friction factor and the water properties at the entered temperature. Every standard size of the chosen material (ASTM B88 type L or K copper, Schedule 40 steel) is tabulated with its velocity and friction rate, and the smallest that meets both the velocity limit and the allowable rate is selected; the residual pressure at the highest fixture follows.

Drainage stack, terminal velocity and vent

The drainage load is expressed in drainage fixture units (BMS 66 Table 803 gives the 1940 values) and converted to a flow by curve 1 of Monograph 31, which is Hunter's flush-valve curve at 10 units per closet. Water entering a stack accelerates down the wall until friction balances gravity; Wyly and Eaton derived the terminal velocity V_t = 3.0 (Q/D)^2/5 ft/s and the terminal length L_t = 0.052 V_t² ft (eqs 46 and 52, Q in gpm, D in inches, cast-iron roughness), one to two stories in practice. The capacity of a tall stack is set by the fraction r_s of the cross section the sheet of water may occupy at terminal velocity, Q = 27.8 r_s^5/3 D^8/3 gpm (eq. 53), with 7/24 the usual design value, 1/4 conservative and 1/3 the upper limit the research supports; their Table 12 is this equation with curve 1. Short stacks are limited by their branches instead: the appendix rule, taken into the 1940 Plumbing Manual, gives a stack of one or two branch intervals the single-interval limit S₁ = 1.5 × the horizontal-branch limit, and a stack of n ≥ 3 intervals S_n = S₁ (n + 2)/4 up to the terminal-velocity capacity, with no one interval loaded beyond S_n/n.

The falling sheet drags air down its core, which the vent must supply within a pressure swing of 1 in of water if the trap seals are to hold. Monograph 31 takes the air flow as Q_a = Q_w (1 − r_s)/r_s (eq. 55 with the air at the water velocity, two-thirds of its upper bound) and the permissible vent length from Darcy-Weisbach at that pressure drop, L = 2200 D_v⁵ / (f Q_a²) ft with D_v the actual inside diameter in inches and f from its Table 13 (0.0367 for a 3-in stack to 0.0214 for 15 in, 0.001 ft roughness); its Table 14 is this equation evaluated for the Schedule 40 vent diameters, which the tool reproduces. The vent must be at least half the stack diameter (Plumbing Manual section 1013).

Building drain

Horizontal drains are sized to run half full at the design flow with the velocity above the self-cleaning value, 2 ft/s being usual. At half depth the area is πD²/8 and the hydraulic radius D/4, so Manning gives Q = (1/n) A R^2/3 S^1/2 (1.486/n in foot units) with n = 0.013 for cast iron; the slope for the self-cleaning velocity is S = (n V / R^2/3)². The tool also finds the normal depth and velocity of the design flow by bisection on the circular section (uniform flow is possible up to 0.938 D) and names the smallest size half full at the entered slope.

Reading the results

Table 1 gives every fixture row alone; the combined demand follows Hunter's rule for the governing curve, and the design flow is the chosen curve plus any continuous demand. Table 2 lists every standard size with its velocity and friction rate by both formulas, the selected size in bold, and Figure 2 draws the friction chart with the allowable rate. The stack section states the capacity in flow and fixture units, the actual fill fraction, the terminal velocity and length, the stack-height limits and whether the load is carried; Table 3 gives the vent length each size allows. The drain section compares the half-full capacity and velocity with the design flow and the self-cleaning value and reports the normal depth.

Limits

Hunter's parameters describe congested public use in 1940 and overstate modern residential demand; enter measured p and q where available. The curves are for intermittently used fixtures; continuous loads are added separately. Stack capacities are for well-vented straight stacks of cast iron or steel of enough height to reach terminal velocity; offsets, interference at stack-branch junctions and the capacity of the horizontal branches themselves are not analysed. The vent formula covers the vent stack alone, not branch vents, circuit vents or wet venting. Hot-water recirculation, water hammer, backflow and the code's minimum sizes for individual fixture branches are outside the tool.