Hardy Cross Pipe Network Calculator

About This Tool

In looped and gridded piping, water reaches a point by more than one route. The flows settle where two conditions hold: the flows into and out of every junction add up, and every route between the same two junctions loses the same pressure. Each pipe’s friction loss follows the Hazen-Williams formula used in NFPA 13, p = 4.52 Q^1.85 / (C^1.85 d^4.87) psi per foot, with Q in gpm and d the inside diameter in inches, applied to the pipe length plus the equivalent length of its fittings. The tool uses the 1.85 exponent of the NFPA 13 form rather than 1.852.

The Hardy Cross method starts from any set of flows that satisfies the junctions, here an even split. For each loop it adds up the friction losses, counting clockwise flow as positive, and applies a flow correction ΔQ = −Σp / (1.85 Σp/Q) around the loop. A pipe shared by two loops takes both corrections. The steps repeat until the losses around every loop cancel. NFPA 13 hand calculations accept a balance within 0.5 psi at hydraulic junctions; the tool reports when that is reached and keeps going to 0.0001 psi. The iteration table shows the loop imbalances shrinking and the flow in a cross-connection changing direction when the first guess was wrong.

For parallel paths between the same two points, the balanced split also has a direct answer: each path carries flow in proportion to (1/R)^0.54, where R is its resistance (the loss at unit flow). The tool solves the paths with Hardy Cross and the result matches that direct split.

Worked example

A grid takes 600 gpm in at corner A and delivers it at the opposite corner D. The outer pipes are 4 in. Schedule 10 (C = 120): 140 ft equivalent on pipes 1 and 4, which turn the corners, and 70 ft on pipes 2 and 5. The cross-connection, pipe 3, is 2-1/2 in. Schedule 40, 50 ft equivalent. Starting from 300 gpm in each outer pipe and nothing in pipe 3, the loops balance within 0.5 psi after three corrections. The result is 329 gpm through pipe 2, 57 gpm down the cross-connection from B to C, 271 gpm through pipes 1 and 4, and a loss of 4.22 psi from A to D by either route.

Assumptions and limits

Flow enters and leaves only at the marked points, the piping is level, and velocity pressure is not included. A real grid with flowing sprinklers has outflows along every branch line and needs a full hydraulic calculation, usually with software. The diameters are for steel Schedule 10 and 40 pipe; equivalent lengths of fittings come from the NFPA 13 table for C = 120 and should be adjusted for other C-factors before they are entered.

When to bring in an engineer

Hand balancing is useful for checking a computer calculation, sizing a loop or a parallel feed, and understanding why a grid behaves the way it does. Sprinkler system calculations for permit are prepared with the actual layout and flowing sprinklers. DGA prepares sprinkler hydraulic calculations and provides third-party review of calculations for gridded and looped systems.

References

1. NFPA 13, Standard for the Installation of Sprinkler Systems, National Fire Protection Association, 2025 edition: hydraulic calculation procedures, the Hazen-Williams formula, equivalent lengths of fittings and valves with C-factor adjustment, and pressure balance at hydraulic junction points.
2. H. Cross, “Analysis of Flow in Networks of Conduits or Conductors,” University of Illinois Engineering Experiment Station, Bulletin No. 286, 1936.
3. G.S. Williams and A. Hazen, Hydraulic Tables, Wiley, New York, 1905.