Friction Loss Formula for Pump Systems: Darcy-Weisbach and Hazen-Williams Worked Examples

Updated August 2026

A friction loss formula is a calculation that converts resistance inside a pipe into head loss or pressure drop. For ordinary water service, the two formulas you will see most often are Darcy-Weisbach and Hazen-Williams. Darcy-Weisbach starts from velocity, pipe geometry, viscosity, and roughness. Hazen-Williams replaces those fluid and surface terms with an empirical C-factor. Both can produce a useful design estimate, but only when the inputs, unit form, and operating range are stated.

This guide works through both methods line by line. It also shows what neither straight-pipe equation includes: valves, fittings, elevation, required discharge pressure, and the pump’s full operating curve. If you already know your inputs and only need a quick numerical result, use BBP’s friction loss calculator. This article owns the method-choice and audit trail; the calculator owns interactive calculation.

Scope: the examples below cover ordinary water service at a declared duty point. They do not establish a shortcut for slurry, pulp stock, non-Newtonian liquids, or an old pipe whose internal condition is unknown. A calculation can be perfectly repeatable and still miss field resistance if the flow, inside diameter, viscosity, roughness, C-factor, or fitting data do not describe the installed line.

This is also not the fire hose coefficient formula that dominates the current search results. A hose friction shortcut is built around different coefficients, units, and operating practice; do not insert its C value into either pipe method below. If you are comparing a friction loss formula in pipe design, a friction loss coefficient, or a friction loss formula Hazen-Williams reference, keep the method’s own coefficient definition and units attached to the calculation.

Which Friction Loss Formula Should You Use?

Use Darcy-Weisbach when the fluid properties, temperature, or pipe condition matter, when the liquid is not plain water, or when you need a method that exposes the flow regime. Use Hazen-Williams for turbulent water service when a defensible C-factor and the correct unit-specific equation are already part of the design basis. Do not choose a formula simply because one calculator asks for fewer inputs.

The U.S. Environmental Protection Agency’s EPANET page lists both Darcy-Weisbach and Hazen-Williams as available friction head-loss methods for water-distribution modeling. That is evidence that both methods have legitimate uses. It is not evidence that the methods share the same assumptions or can be swapped after a calculation has started.

A useful selection rule is simple: if you can’t explain where the C-factor came from, use Darcy-Weisbach and document roughness, viscosity, and the friction-factor method instead. If you can’t classify the Reynolds-number regime, neither method should be treated as a push-button answer.

Darcy-Weisbach Equation: Inputs, Friction Factor, and Units

One common mistake is to treat one friction factor as fixed across every flow. That can produce a wrong pressure-drop result because Reynolds number and relative roughness change the resistance model.

The Darcy-Weisbach equation calculates major head loss along a constant-diameter pipe. In head form, it is:

Here, h_f is friction head loss in meters or feet of the flowing liquid; f is the dimensionless Darcy friction factor; L is straight-pipe length; D is actual inside diameter; v is mean velocity; and g is gravitational acceleration. Keep L and D in the same length unit. Head then comes out in that unit.

The diameter of the pipe and the length of pipe set the geometry; the viscosity of the fluid, the pipe surface condition, and turbulence determine resistance. Record the internal diameter, Schedule 40 or another exact wall schedule, and the loss of pressure at each design flow. A Moody diagram can provide a theoretical friction factor, but an accurate and efficient pump selection still depends on the actual supply-line condition.

Velocity comes from the actual flow area, not the nominal pipe label:

Next, classify the flow with Reynolds number:

The Hydraulic Institute pipe-friction reference places laminar flow below about Re = 2,000, turbulent flow above about Re = 4,000, and calls the interval between them a critical or transition zone. For laminar flow, f = 64/Re. For turbulent flow, use a documented Colebrook solution, Moody chart, or named explicit approximation. Do not feed a transition-zone Reynolds number into a turbulent correlation and report the result as uniquely determined.

“The Colebrook Equation offers a reliable means for computing the Darcy-Weisbach friction factor.”

For a turbulent example that must be reproducible without an iterative solver, this article uses the Swamee-Jain approximation:

That choice is part of the record. Someone repeating the work with an iterative Colebrook solution may obtain a slightly different last decimal. That is acceptable; hiding the selected friction-factor convention is not.

Darcy-Weisbach Worked Example in SI Units

Suppose water at approximately 20°C flows through 100 m of commercial-steel pipe. The declared inputs are Q = 0.010 m³/s, D = 0.100 m actual inside diameter, absolute roughness ε = 0.000045 m, and assumed kinematic viscosity ν = 1.004 × 10^-6 m²/s. The example ignores fittings and elevation so that it isolates major loss under the Hydraulic Institute major-loss formulation.

1. Calculate area: A = π(0.100)²/4 = 0.007854 m².
2. Calculate velocity: v = 0.010/0.007854 = 1.273 m/s.
3. Calculate Reynolds number: Re = (1.273 × 0.100)/(1.004 × 10^-6) = 126,817. This is well above 4,000, so the turbulent method is appropriate.
4. Calculate Darcy friction factor: the stated Swamee-Jain form gives f = 0.01960.
5. Calculate head loss: h_f = 0.01960 × (100/0.100) × [1.273²/(2 × 9.80665)] = 1.620 m.
6. Convert head to pressure for water: using assumed ρ = 1,000 kg/m³, Δp = ρgh = 1,000 × 9.80665 × 1.620 = 15.89 kPa.

The arithmetic is auditable, but the field result is only as sound as the inputs. A 5% diameter assumption error can create a much larger head-loss error because velocity and L/D both change. If the pipe has deposits, corrosion, a different schedule, or a temperature that changes viscosity, rerun the calculation with measured or project-approved values. “Commercial steel” is not a substitute for inspecting an old line.

Hazen-Williams Formula: Where the C-Factor Fits

Hazen-Williams is an empirical water-pipe formula. In one common U.S. customary pressure-loss form, with pressure loss in psi, length in feet, flow in U.S. gallons per minute, and inside diameter in inches, it is:

The constant 4.52 belongs to that exact unit set and pressure form. A metric head-loss version uses a different constant. Copying the constant while changing gallons per minute to cubic meters per hour, inches to millimeters, or psi to meters of head will not produce a unit conversion; it will produce a wrong answer.

The C-factor is not a measured wall height like Darcy roughness. It is an empirical resistance coefficient tied to pipe material, condition, diameter, and the data behind the selected value. An attributed comparison published on ASPE Pipeline notes that Hazen-Williams does not contain temperature, density, or viscosity terms. That is why this article keeps its use to turbulent water service and treats C as a design assumption.

3-Level Coefficient-Provenance Framework

A green provenance record does not promise that the installed line matches the coefficient forever. It means the assumption is traceable and was suitable for the stated design decision. Field calibration remains the stronger basis when aging, deposits, or operating history can change resistance.

Hazen-Williams Worked Example in US Customary Units

Consider 500 ft of water pipe carrying 500 gpm. The actual inside diameter is 6.065 in and the declared Hazen-Williams C-factor is 130. The NIST SI conversion table lists 1 U.S. gpm as 6.309020 × 10^-5 m³/s; retaining that conversion in the receipt makes the cross-check reproducible.

1. Insert the stated values: Δp = 4.52 × 500 × 500^1.852 / [130^1.852 × 6.065^4.87].
2. Calculate pressure loss: Δp = 4.219 psi across 500 ft.
3. Normalize by length: 4.219/5 = 0.844 psi per 100 ft.
4. Convert to water head: 4.219 psi corresponds to about 9.73 ft of water head at the stated water condition.

This example is also turbulent. Using the same line’s flow and diameter with ordinary water gives Re ≈ 230,599, comfortably above the transition zone. That check matters because “water service” alone is not a complete Hazen-Williams boundary.

Darcy-Weisbach vs Hazen-Williams: Same-Line Reconciliation

The practical risk is a pump-selection mismatch: two valid-looking methods can disagree because their resistance inputs represent different pipe assumptions. The U.S. Environmental Protection Agency’s EPANET method list keeps Darcy-Weisbach and Hazen-Williams as distinct choices, so treat the gap as a review trigger, not a reason to average the answers.

Now apply Darcy-Weisbach to the same 500 gpm, 6.065 in ID, 500 ft water line. Declare commercial-steel roughness as 0.00015 ft and assumed water kinematic viscosity as 1.217 × 10^-5 ft²/s. The calculation gives velocity 5.553 ft/s, Re = 230,599, Darcy f = 0.01749, and head loss = 8.292 ft. That equals about 3.594 psi for water.

The 17.4% gap is not a universal correction factor. It is the result of pairing one roughness assumption with one C-factor at one duty point. Other valid water-distribution inputs may make the methods agree more closely or diverge further. The table’s purpose is to expose the modeling assumptions, not crown a winner from one line.

When the two answers differ enough to affect pump selection, pipe size, or energy cost, do not average them. Check actual inside diameter, flow measurement, temperature, roughness or C-factor origin, and flow-regime method. Then choose the equation that matches the project’s design basis.

Add Minor Losses Before You Use Friction Loss in TDH

Darcy-Weisbach major loss covers wall friction in straight pipe. A valve, elbow, tee, reducer, strainer, entrance, or exit adds a local loss. The Hydraulic Institute fitting reference expresses one component as:

If all K values use the same reference velocity and compatible definitions, major and minor losses can be combined:

The Hydraulic Institute fitting-loss page also shows why K is not a magic fixed number. Published ranges vary with fitting type, geometry, size, and condition; the page’s examples range from roughly ±10% for some 45-degree elbows to +200%/-80% for a flanged check valve. Record valve position and fitting geometry rather than copying a bare K column.

Total dynamic head goes one step further:

Do not add nozzle pressure, static head, or fitting losses inside the straight-pipe formula and then add them again in TDH. That double-counting produces an oversized duty point. Use BBP’s total dynamic head calculator when you are ready to assemble the complete head balance.

Why Flow, Diameter, and Roughness Change the Result

At fixed flow, a smaller inside diameter raises velocity and increases the length-to-diameter ratio at the same time. That makes diameter errors far more damaging than their percentage suggests. Recomputing the 500 gpm Darcy line at three actual diameters gives:

This is the Head-to-Pressure Dimensional Checksum: confirm flow unit, confirm actual diameter, compute velocity, classify Reynolds number, select the resistance method, calculate head, and only then convert head to pressure for a named fluid. Reversing that order often hides a unit error.

One duty-point result still does not describe the full system curve. Flow changes velocity, Reynolds number, friction factor, and loss. The 2026 Hydraulic Institute data-tool overview treats system curves, pump curves, and their intersection as separate parts of pump-system analysis. Calculate enough points to locate the expected operating range, then compare that curve with the selected pump curve.

Once the duty head is established, connect it to motor demand with BBP’s pump power formula. Friction head raises required pump head; required head and flow then determine hydraulic power.

Friction Loss Calculation Audit Checklist

A calculation record should let another engineer reproduce the result without guessing what “standard pipe” or “normal water” meant. Save the following items with the pump request for quotation or design note:

• Fluid name, concentration if relevant, temperature, density, and viscosity source.
• Flow rate, unit, operating range, and whether it was measured or specified.
• Pipe material, schedule, actual inside diameter, length by diameter, age, lining, and known deposits.
• Selected formula and exact unit form.
• Reynolds number and the flow-regime decision.
• Darcy friction factor method plus roughness source, or Hazen-Williams C-factor plus provenance.
• Every fitting and valve, its position, K or equivalent-length source, and reference velocity.
• Major loss, minor loss, static head, required endpoint pressure, and final TDH shown separately.
• Head-to-pressure conversion with named fluid density.
• Known exclusions, uncertainty, revision date, and calculation owner.

Send this record with your request for quotation to compare the duty point with the relevant centrifugal pump range or irrigation pump system. For slurry or another specialized fluid, provide the fluid data and application details rather than substituting the clean-water examples above.

Frequently Asked Questions