Pipe Flow & Head Loss on the PE WRE Exam: Hazen-Williams, Darcy, and Minor Losses

Pipe-flow problems are the meat of the PE Civil WRE Hydraulics — Closed Conduit block, and two equations dominate: Hazen–Williams (water at typical temperatures, simpler form, water-utility default) and Darcy–Weisbach (more general, requires the Moody diagram or Colebrook iteration to find f). The reader pain isn't formula manipulation — both forms are reproduced in the handbook. The pain is knowing when to use which: Hazen–Williams for water at 40–75°F flowing in pipes that aren't too small or too rough; Darcy–Weisbach for everything else, especially gas, oil, water at non-standard temperatures, or when the problem gives you pipe roughness.

The other persistent error is minor losses. The big head-loss number is friction loss along the pipe, but the K-factors for entrances, exits, fittings, valves, and bends can collectively eat 20–40% of the friction loss on short systems. Forgetting them in a pump-sizing problem leads to under-sized pumps that don't deliver design flow.

This post walks through the four pipe-flow problem types NCEES tests, with three fully solved worked examples (Hazen–Williams head loss, Darcy–Weisbach with Moody diagram, parallel-pipe head-loss split) plus reference tables for Hazen–Williams C values and minor-loss K-factors. Every formula traces to its section in the NCEES PE Civil Reference Handbook (§6.3, Closed Conduit Flow and Pumps).

Why pipe flow matters on the PE WRE exam

Per the April 2024 NCEES PE Civil WRE specification, Topic 5 (Hydraulics — Closed Conduit) carries 7–11 questions out of 80, with sub-topic 5B (pressure conduit) covering single pipe, force mains, Hazen–Williams, Darcy–Weisbach, and major and minor losses. Pipe flow also underpins sub-topic 5C (pump application and analysis) and 5D (pipe network analysis — series, parallel, and loop networks), and feeds Topic 10 (Drinking Water Distribution and Treatment) and Topic 11 (Wastewater Collection and Treatment, including force mains). A confident command of head-loss calculations is upstream of nearly every pressurized-system question on the exam.

Core concepts you must master

Hazen–Williams equation

Empirical, calibrated for water at typical pipe-utility temperatures. The handbook gives a head-loss-in-feet form (§6.3.1.2):

where h_f = friction head loss (ft), L = pipe length (ft), Q = flow rate (cfs), C = Hazen–Williams coefficient (dimensionless), D = inside diameter (ft). The handbook also lists a pressure form (§6.3.1.3) with constant 4.52 that takes Q in gpm and D in inches and returns psi per foot of pipe — don't mix the two; this post uses the feet form throughout. Equivalent velocity form: V = 1.318·C·R^0.63·S^0.54, where R = hydraulic radius (= D/4 for full pipe), S = slope of energy grade line (= h_f/L).

Validity: water at typical temperatures (40–75°F), V < ~10 ft/s, D > 2 in. Outside this range, Darcy–Weisbach is the right tool.

Darcy–Weisbach equation

Theoretically grounded, valid for any Newtonian fluid at any temperature:

where f = Darcy friction factor (dimensionless), found from the Moody diagram given Reynolds number Re = VD/ν and relative roughness ε/D. For smooth turbulent flow, the Colebrook equation gives f implicitly:

For the exam, read f directly off the Moody diagram in the handbook — the Colebrook iteration is too slow for a 6-minute question.

Minor losses (K-factor method)

Fittings, valves, entrances, exits, and bends each dissipate some velocity head:

K values for common fittings (handbook §6.3.3, which uses the symbol C for this coefficient): sharp-edge entrance 0.5; long-radius 90° elbow 0.6; gate valve fully open 0.2; globe valve 10; exit to reservoir 1.0; sudden contraction varies with area ratio. Sum the K values and apply once with the appropriate velocity. Equivalent-length method reframes K as L_eq = K·D/f, which is convenient when adding up to a long pipe but obscures the velocity dependence.

Parallel and series pipe networks

Two pipes in series: same flow Q, head losses add (h_total = h₁ + h₂). Two pipes in parallel: same head loss across both branches (h₁ = h₂), flows add (Q_total = Q₁ + Q₂). For Hazen–Williams parallel pipes with L₁ = L₂, the flow split simplifies to Q₁/Q₂ = (C₁/C₂)·(D₁/D₂)^2.63. Worked below.

System head and pump head

For a pumped system: pump-supplied head = static lift + total friction loss + total minor loss + velocity head at outlet (if discharging to atmosphere) − inlet head. The pump operating point is where this system curve intersects the pump's H-Q curve. (Detailed pump-side analysis is in our pump hydraulics post; this post is purely about the friction/minor-loss side.)

The 4 types of pipe-flow problems on the PE WRE exam

Type 1: Hazen–Williams head loss

Given pipe length, diameter, Hazen–Williams C, and flow rate. Compute h_f directly from the equation. Watch units — the feet form (constant 4.73) takes Q in cfs and D in ft, so convert from gpm and inches first; the separate pressure form (4.52, gpm, inches) returns psi per foot, not feet. Worked below.

Type 2: Darcy–Weisbach with Moody diagram

Given pipe geometry, water properties (kinematic viscosity at the given temperature), and flow rate. Compute V, then Re, then ε/D, then read f from the Moody chart. Apply h_f = f·(L/D)·V²/(2g). Worked below.

Type 3: Minor losses (fittings, valves, entrances/exits)

Given a pipe with a list of fittings (entrances, elbows, valves, exits) and a flow rate. Sum the K-factors, compute h_m = (ΣK)·V²/(2g). Add to the friction h_f for total head loss.

Type 4: Parallel-pipe head-loss split

Given two pipes connecting the same two nodes and a total flow Q, find Q₁ and Q₂. Same head loss across both branches sets the constraint. For Hazen–Williams the closed-form ratio above gives the split directly. Worked below.

Worked example: Hazen–Williams head loss

Worked example: Darcy–Weisbach with Moody diagram

Worked example: parallel-pipe head-loss split

Common errors that cost points

Using Hazen–Williams outside its valid range

Hazen–Williams is calibrated for water at 40–75°F flowing at moderate velocities in pipes ≥ 2 in diameter. For non-water fluids, water at higher or lower temperatures, very small pipes, or very rough pipes, Darcy–Weisbach is the right tool. Using HW for a 60°F oil pipeline gives nonsense. Read the question for fluid identity and conditions.

Forgetting the velocity head term

Total energy at any point = pressure head + elevation head + velocity head (V²/(2g)). When tracking energy from one cross-section to another, the velocity head changes if the pipe diameter changes. For most water-utility scenarios velocity heads are 0.05–1 ft (small relative to friction loss over long runs), but for short systems with diameter changes the velocity-head difference can dominate the energy balance.

Wrong K-factor for fittings

K-factor tables vary across references. 90° elbow: 0.6 (long-radius) to 0.9 (short-radius). Globe valve fully open: 10 in the handbook (6 to 12 across other references). Check the source the exam reference uses (handbook §6.3.3 reproduces a loss-coefficient table). Using a K from a different reference can throw minor losses off by 50%.

Ignoring water-temperature effect on viscosity for Darcy–Weisbach

Kinematic viscosity of water varies from 1.93 × 10⁻⁵ ft²/s at 32°F to 0.74 × 10⁻⁵ ft²/s at 100°F — almost a factor of 3. Reynolds number scales inversely with viscosity, so cold water in the same pipe has Re three times lower than hot water, which can shift the Moody-chart f-value by 10–20%. The handbook's water-property tables give ν at 60°F (1.21 × 10⁻⁵) as a default; if the question specifies a different temperature, use the right ν.

How to study pipe flow for the PE WRE exam

Phase 1 — Equation fluency (Week 1)

Read handbook §6.3 (closed conduit flow and pumps) end-to-end. Practice writing both Hazen–Williams and Darcy–Weisbach from a blank page until you can pick the right equation in under 30 seconds based on fluid type, temperature, and what the question provides.

Phase 2 — Worked-problem drills (Weeks 2–3)

Work twelve problems across the four types: four Hazen–Williams (vary pipe size, length, C); four Darcy–Weisbach with Moody (vary Re and ε/D); two with significant minor losses; two parallel-pipe splits. Time yourself: three to five minutes per problem on the exam. PEwise's Module 7 (Flow in Pipe Systems) covers Hazen–Williams, Darcy–Weisbach with Moody diagram, minor-loss K-factors, and parallel/series pipe network analysis in 18 lessons with worked NCEES-style problems.

Phase 3 — Integration with pump systems and water distribution (Week 4)

Solve five integration problems where you start from a pumped system layout, sum friction + minor losses across the system to build a system head curve, and find the pump operating point at the intersection of the pump and system curves. That chain (pipe friction + minor losses → system head → pump operating point) is the realistic Topic-5 + Topic-4 pattern on the exam.

Quick reference: Hazen–Williams C and minor-loss K

Hazen–Williams C values (typical)

Sources: NCEES PE Civil Reference Handbook §6.3.1; AWWA C150/C151; Mays (Water Distribution Systems Handbook).

Minor-loss K-factors (typical)

Sources: NCEES PE Civil Reference Handbook §6.3.3 (fitting values shown as loss coefficient C); Crane Technical Paper No. 410. Use the values your exam reference specifies.

Connecting this to your overall PE WRE exam strategy

Pipe flow sits upstream of pump-system analysis (sum friction + minor losses to build the system head curve) and water-distribution network design (Hardy Cross or Newton-Raphson on a multi-loop network). Get the head-loss equation choice automatic, and the pump-sizing and network problems collapse to a sequence of clean steps. For the pump-system side that uses these head-loss results to size pumps, see our PE WRE pump hydraulics post. For the open-channel counterpart (gravity flow rather than pressurized), see the open-channel-flow problem-types post.

Final thoughts

Pipe-flow problems reward engineers who treat equation selection as the first 30 seconds of the question. Water at typical temperatures and the question gives a Hazen–Williams C? Use HW directly. Non-water fluid, non-standard temperature, or the question gives roughness ε? Use DW with the Moody diagram. Once the equation is fixed, the calculation is mechanical. Drill the equation-selection check until it's automatic.