Hazen-Williams, pipe networks and pumps on the PE Civil WRE exam

Pressurized water systems, such as distribution mains, force mains, booster and lift stations, and fire flow, share one workflow. You compute the head loss in each pipe, combine the pipes, then match the system to a pump. The Handbook's Closed Conduit Flow and Pumps section holds the relations for every step except the loop correction.

What you actually need to own

• Hazen-Williams is empirical. It applies to water in turbulent flow at ordinary temperatures, and roughness enters as C, where a higher C means a smoother pipe. In feet and cfs: Q = 0.432·C·D^2.63·S^0.54 and h_f = 4.73·L·Q^1.852 / (C^1.852·D^4.87), with D in feet. The pressure-per-foot form uses gpm and D in inches, with a different constant.
• Darcy-Weisbach, h_f = f(L/D)(V²/2g), works for any fluid. It needs Re and ε/D to get f from the Moody diagram. Loss goes as roughly Q² and D^−5. Use whichever method the problem sets up: a C value points to Hazen-Williams, and a roughness with a viscosity points to Darcy. The two methods do not give identical answers.
• Series pipes carry the same Q, and their losses add. Parallel pipes have the same head loss, and their flows add. With Hazen-Williams at equal h_f, each branch carries flow in proportion to C·D^2.63 / L^0.54. That ratio splits a total flow without trial and error.
• Loops (Hardy Cross). Assume flows that satisfy continuity at every node. Then correct each loop by ΔQ = −Σh_f / (n·Σ|h_f/Q|), with n = 1.85 for Hazen-Williams or 2 for Darcy. The Handbook prints network continuity and equal loss in parallel pipes, but not this correction, so memorize it.
• System curve and operating point. The system curve is H = static lift + losses, and the losses scale as Q^1.85 or Q². The pump runs where its curve crosses the system curve.
• Pumps in series add head at the same flow. Pumps in parallel add flow at the same head.
• Power. Water horsepower = γQH/550, with Q in cfs, or Q(gpm)·H(ft)/3,956 for water (often rounded to 3,960). Divide by pump efficiency to get brake power, and by pump × motor efficiency to get input power.
• NPSHA = atmospheric head + static suction head − suction-line losses − vapor-pressure head, all in feet of the liquid. Static suction head is negative for a suction lift. NPSHA must exceed the pump's NPSHR.

Where people lose points

• Mixing Hazen-Williams unit sets. Putting D in inches into the feet-and-cfs form, or Q in gpm into it.
• Scaling a loss linearly with flow. A loss quoted at one flow is not a fixed head at another flow. Scale it by (Q₂/Q₁)^1.85 or squared.
• Averaging diameters for parallel or series pipes. Pair each diameter with its own length. The Handbook's parallel-pipe line with D², D_A² and D_B² is flow continuity, area times velocity in each branch, so each branch keeps its own velocity.
• Expecting two parallel pumps to double the flow. The new operating point climbs the system curve, so the combined flow falls further short of double as the system curve steepens. Parallel pumps gain most on a flat, static-dominated system curve. Series pumps gain most on a steep, friction-dominated one.
• Scaling the operating point with the affinity laws when the system has static lift. The laws map each pump-curve point along a parabola through the origin (H ∝ Q²). Scale the whole pump curve to the new speed, then find where it crosses the system curve. Q ∝ N holds for the operating point only on a pure-friction system curve.
• NPSHA errors. Wrong sign on a suction lift, vapor pressure read at the wrong water temperature, or sea-level atmospheric head used at an elevated site.
• Loop signs. Each pipe takes its sign from its direction around the loop, and a pipe shared by two loops takes opposite signs in each.

How to study it

Set up a small template: one Hazen-Williams loss in each unit set, one parallel split, one Hardy Cross iteration, and one operating point with an NPSHA check. Rework it until the unit bookkeeping stops costing you time. When you practice, CastorPrep's tutor works through the problem in front of you. Gravity sewers flowing partly full use Manning's equation instead, and distribution-system pumping connects to treatment through CT, sedimentation and filtration.

Part of the Hydraulics–Closed Conduit area of the PE Civil WRE exam. → Start practicing free — the whole practice bank, free with an account.