Worked problem

Sprinkler Branch Line: Work Back from the Remote Head

Work back from a 15 gpm remote sprinkler through three heads, friction and a 4 ft rise, letting each head flow more as its pressure rises.

Fire Protection · Last reviewed October 17, 2026

Given

A branch line with three sprinklers, each with K = 5.6 gpm/psi^½, spaced 10 ft apart. The most remote head (head 1) must flow 15 gpm. Pipe is Schedule 40 steel with Hazen–Williams C = 120: 10 ft of 1 in. (inside diameter 1.049 in.) from head 1 to head 2, 10 ft of 1¼ in. (1.380 in.) from head 2 to head 3, and then 6 ft of 1½ in. (1.610 in.) up to the cross main, rising 4 ft. Ignore fittings for this simplified exercise.

Find

The pressure and flow at each head, and the pressure and flow at the cross-main connection.

Solution

Hazen–Williams: p (psi/ft) = 4.52 × Q^1.85 ÷ (C^1.85 × d^4.87)     Sprinkler: Q = K × √P     Elevation: 0.433 psi/ft

Head 1: P1 = (Q ÷ K)² = (15 ÷ 5.6)² = 7.17 psi.
Friction from head 1 to head 2 (15 gpm, 1 in.): 0.76 psi. So P2 = 7.17 + 0.76 = 7.94 psi.
Head 2 discharges Q2 = 5.6 × √7.94 = 15.78 gpm. The flow in the next pipe is 15.00 + 15.78 = 30.78 gpm.
Friction from head 2 to head 3 (30.78 gpm, 1¼ in.): 0.76 psi. So P3 = 7.94 + 0.76 = 8.70 psi.
Head 3 discharges Q3 = 5.6 × √8.70 = 16.52 gpm. The branch total is 30.78 + 16.52 = 47.30 gpm.
Last pipe (47.30 gpm, 1½ in., 6 ft): friction 0.48 psi, plus elevation 4 ft × 0.433 = 1.73 psi. The pressure at the cross main = 8.70 + 0.48 + 1.73 = 10.91 psi.

Answer

Point	Pressure	Flow from the head	Flow in pipe after it
Head 1 (remote)	7.17 psi	15.00 gpm	15.00 gpm
Head 2	7.94 psi	15.78 gpm	30.78 gpm
Head 3	8.70 psi	16.52 gpm	47.30 gpm
Cross-main connection	10.91 psi		47.30 gpm

branch demand: 47.3 gpm at 10.9 psi at the cross-main connection

Why the flows are not all 15 gpm

A sprinkler discharges in proportion to the square root of the pressure at its inlet. Heads closer to the supply see a higher pressure, so they discharge more. If every head were assumed to flow 15 gpm, the demand would be 45 gpm and the pressure 10.83 psi, which is 5% too little flow. The error grows with longer branch lines and larger pressure differences.

Common mistakes

Working from the supply downstream with a guessed pressure, instead of from the remote head upstream, which gives a direct answer with no iteration.
Using the flow of the head alone in each pipe segment, instead of the cumulative flow downstream of that point.
Leaving out the elevation pressure for a rise, or counting it with the wrong sign for a drop.
Using the nominal pipe diameter in the friction formula instead of the actual inside diameter.

A real hydraulic calculation also includes fittings as equivalent length, the code minimums for pressure and density, and the area of coverage. This is a simplified exercise.

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

Questions

Why is the remote head the starting point?

It is the most demanding head, with the lowest pressure. Starting there fixes the flow and pressure at one end, and each pipe segment then adds friction and flow going back to the supply.

What does K = 5.6 mean?

It is the discharge coefficient of the sprinkler orifice in gpm per square root of psi. A head at 10 psi flows 5.6 × √10 = 17.7 gpm.

Is 15 gpm a code number?

It is an exercise value. The required flow comes from the design density and the area each sprinkler covers.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

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