Sprinkler Hydraulics

Sprinkler Hydraulics

Sprinkler Hydraulics

A hydraulic calculation shows that the water supply can feed the sprinklers of the hydraulically most demanding area at their design density, with the hose streams the fire department will add, through the pipe as laid out. The tool reproduces the calculation sheet that a designer fills in by hand or with a listed program: it starts at the most remote head, works back to the supply one pipe at a time, balances the side branches at every junction, and sets the demand point against the supply curve of a hydrant flow test. A looped (gridded) system, which has no single path, is balanced by the Hardy Cross method instead. Both are checked by an exact solution of the network equations printed beside them.

Units and the sheet

The calculation runs in the units of a sprinkler sheet: flow Q in gallons per minute, pressure p in pounds per square inch, lengths in feet and inside diameters in inches, with the K-factor in gpm per √(lb/in²). In SI mode every entry and result is converted (L/s, kPa, m, mm) and the metric K (L/min per √bar, 14.4 times the US value) is printed; the formulas below are the ones the sheet uses. One metre of water is 9.81 kPa; one foot is 0.433 lb/in².

Design criteria

The design density and the design area come from the hazard classification: NFPA 13 gives them as a density–area curve for each hazard, which is the standard's own table and is not reproduced here; read the pair for the hazard from the standard (or from the owner's insurer) and enter it. The defaults are the light-hazard values that UFC 3-600-01 Table 9-3 prints for wet systems under ceilings up to 30 ft, 0.10 gpm/ft² over 1,500 ft² with a minimum K of 5.6, and its Table 9-4 hose allowance of 250 gpm for 60 minutes. The same table gives ordinary hazard 0.20 gpm/ft² over 2,500 ft² (K 8.0) and extra hazard 0.30 gpm/ft² over 2,500 ft² (K 11.2), and raises the area for dry systems and high ceilings.

The number of flowing heads is the design area divided by the coverage of one head, rounded up, and the number on one branch line is 1.2 √(design area) / spacing, rounded up, so that the design area is a rectangle longer along the branch lines than across them. The heads listed in the pipe table are the flowing heads; heads outside the design area are left out, and the pipes that only serve them are left out too. Every flowing head must discharge at least density × coverage area; by Q = K √p that is a pressure of (density × area / K)², and the larger of this and the entered minimum head pressure (7 lb/in² for a standard spray sprinkler) is used at every end head.

Discharge, friction, elevation and fittings

A sprinkler discharges Q = K √p. The friction loss per unit length of pipe is the Hazen–Williams formula in the sprinkler form,

p_f = 4.52 Q^1.85 / (C^1.85 d^4.87) lb/in² per ft (Q in gpm, d in inches), or p_f = 6.05 × 10⁵ Q^1.85 / (C^1.85 d^4.87) bar per m (Q in L/min, d in mm),

which is Williams and Hazen's V = 1.318 C R^0.63 S^0.54 written for these units with the exponents rounded; the rounding and the constant 4.52 (the exact conversion gives 4.54) keep the sheet form within 0.3 % of the formula at sprinkler flows and about 1.5 % at 9,000 gpm. C is the roughness coefficient of the pipe: 120 for steel in a wet system, 100 in a dry system, 140 for cement-lined ductile iron, 150 for copper or plastic. The velocity is v = 0.4085 Q / d² ft/s. The elevation term is 0.433 lb/in² per foot of rise between the two ends of the pipe. Devices with a listed pressure drop at the design flow (backflow preventers, alarm or dry-pipe valves, flow meters) are entered per pipe as a fixed loss; UFC 3-600-01 asks for at least 8 lb/in² across a double check assembly and 12 across a reduced-pressure assembly.

Fittings are entered by count and code and charged as a loss coefficient K times the velocity head, K v²/(2g), with the coefficients of NRCS Engineering Field Handbook chapter 3 Table 9 as defaults (standard elbow 0.90, long-radius elbow 0.60, 45° elbow 0.42, tee or cross through the side outlet 1.80, gate valve 0.19); butterfly, check and alarm valves have no public-domain coefficient and take the listed value. The sheet prints each group of fittings as the equivalent length of the pipe it sits in, L_eq = K v²/(2g) × 0.433 / p_f, evaluated at the pipe's own flow, so that the total length column reads as it does on a hand sheet; because both the fitting loss and the friction loss are powers of the flow, this equivalent length grows as d^0.87 with the pipe size and only as Q^0.15 with the flow. An analyst who prefers the equivalent-length table of the standard enters its figure in the extra-length column instead.

The node method on a tree

Each pipe row names its downstream node (node 1, where a head and an elevation may sit) and its upstream node (node 2, toward the supply). Starting at an end head at its minimum pressure, the discharge is K √p; the pipe to the next node loses p_f × total length plus the elevation and device terms, which gives the pressure at that node; the head there discharges K √p of that pressure and the flows add; and so on to the supply. Where a side branch joins, that branch has first been computed on its own from its own end head at the minimum, which gives the flow and pressure it needs at the junction and hence its K_eq = Q / √p; the branch needing the higher pressure governs the junction, and the other flows K_eq √p of the governing pressure, its heads scaled up with it. The tool applies this recursion to any tree, so the governing path, and with it the hydraulically most remote head, comes out of the calculation rather than being chosen. The hose allowance enters at the node named for it (the supply point, or the riser base for inside hose connections) and is carried by the pipes between that node and the supply.

The K_eq step assumes the branch's pressure grows as Q², while the pipe loss in it grows as Q^1.85; a scaled branch therefore carries a little more water than the sheet gives it. The exact solution shows the size of that difference, a fraction of a percent on the default system.

Gridded systems: Hardy Cross

A gridded system feeds every branch line from both ends, so each flowing head has several paths to the supply and the sheet method cannot start anywhere. The tool detects the loops (pipes minus nodes plus one) and balances the flows by Hardy Cross: with the head flows fixed, a spanning tree gives flows that satisfy continuity, and the chords close the fundamental loops; around each loop the signed losses must sum to zero, and the correction ΔQ = −Σh / Σ|dh/dQ| is applied to every pipe of the loop until the corrections vanish, with h = sign(Q)(4.52 L Q^1.85/(C^1.85 d^4.87) + ΣK v²/(2g) × 0.433 + device). The node pressures then follow along the spanning tree, the lowest head is held at its minimum, every head discharges K √p of its pressure, and the balance is repeated with the new head flows until they settle: the classical procedure of the standards for gridded systems, in which the remote head is found by the iteration itself.

The exact network solution

Beside either method the tool solves the network as a set of nodal equations by Newton's method: unknown pressures at every node but the reference head, which is held at its minimum, and an unknown inflow at the supply node; every head discharges K √p, every pipe carries the flow that its own pressure difference drives through the Hazen–Williams law with the fittings and devices included, and continuity holds at every node. If another head would fall below its minimum the reference moves to it and the solve repeats. For a tree this is the sheet method without the K_eq approximation; for a grid it is the Hardy Cross result to machine precision. The nodal table and the pipe table show both.

Demand against the supply

The demand point is the total flow (heads plus hose) at the pressure the sheet needs at the supply node. The supply is a hydrant flow test: static pressure p_s, residual pressure p_r at the test flow Q_r. Because the loss in the mains follows the same Q^1.85 law, the available pressure at any flow is p = p_s − (p_s − p_r)(Q / Q_r)^1.85, a straight line on N^1.85 graph paper, and the flow available at the required pressure follows by inverting it. The supply is adequate when the available pressure at the demand flow is at or above the required pressure; the margin is printed as a plain difference, and when it is negative the tool gives the boost a fire pump must add at the demand flow. The charts draw the supply curve, the demand with its hose segment, the hydrant test point and a system curve through the demand point (elevation plus a Q^1.85 friction part), on a linear flow axis and on the N^1.85 axis.

Checks and limits

The sheet form of Hazen–Williams is checked against the formula, against the NRCS handbook's worked pipe examples and against Darcy–Weisbach through the Colebrook equation (printed for the remote pipe); the tree walk, the exact solution and the Hardy Cross balance are checked against independent solutions of the same networks. Not covered: velocity pressure (the standard's optional normal-pressure method, which lowers the discharge of heads on fast branch lines), the pipe-schedule method, dry-system water delivery time, the water-supply duration and tank sizing, and the sizing of the fire pump beyond the boost it must give. Inside diameters are those of ASME B36.10M Schedule 40 and Schedule 10 steel; any other pipe takes its inside diameter on its row. The NFPA 13 tables of densities, areas, hose allowances and fitting equivalent lengths are the standard's and are not reproduced: the tool takes those as entered values and names the clauses.