Darcy-Weisbach vs Hazen-Williams: Comparing Pipe Flow Methods

Quick Comparison Overview

The Darcy-Weisbach Equation

Julius Weisbach formulated the basis of this equation in 1845 and Henry Darcy extended it with experimental pipe data through the 1850s. The result is a physically derived relationship that holds across all flow conditions and fluid types. It is the standard method in mechanical, chemical, oil and gas, and aerospace engineering, and is increasingly the standard in civil engineering water system design as well.

Head Loss Form

Pressure Drop Form

You can also express the equation in terms of pressure drop rather than head loss. The two forms are equivalent through ΔP = ρgh_f:

Use the pressure form when working in SI units with pumps specified in pascals or bar. Use the head form when working with elevation differences and pump head in metres.

What Darcy-Weisbach Requires

To use the equation, you need the Darcy friction factor f, which itself requires Reynolds number and relative roughness. That means you also need fluid density and viscosity at operating temperature, pipe inner diameter, and absolute roughness for the pipe material. The Moody Chart Calculator handles the friction factor step. For roughness values by material, see the pipe roughness guide. For Reynolds number calculation, see the Reynolds number guide.

Strengths of Darcy-Weisbach

The equation works for any Newtonian fluid: water, oil, air, gas, solvents, refrigerants. It works at any temperature as long as you use the correct viscosity and density. It covers laminar, transitional, and turbulent flow. It works for any pipe diameter and any velocity. It is dimensionally consistent, so you get exact results with consistent SI or Imperial units. It accounts for both Reynolds-number effects and roughness effects through the friction factor.

The only reason engineers use anything else is that finding the friction factor requires either reading the Moody Chart or solving the Colebrook-White equation. The Swamee-Jain equation eliminates the iteration for quick hand calculations, making Darcy-Weisbach practical in all situations.

The Hazen-Williams Equation

Allen Hazen and Gardner Williams published this formula in 1902 based on empirical data from water mains in the United States. Its popularity in water supply engineering comes from its simplicity: you do not need to calculate Reynolds number or friction factor. You look up a single coefficient C for your pipe material and get velocity directly.

Velocity Form (SI)

Head Loss Form

Rearranging for head loss and substituting R = D/4 for circular pipes:

Where Q is volumetric flow rate in m³/s and D is inner diameter in metres. This is the form most used in hydraulic network analysis software like EPANET.

US Customary Form

In US practice, Hazen-Williams is often expressed with Q in gallons per minute and D in inches:

The numerical coefficient changes but the structure is identical. Make sure you know which unit system a given reference uses before applying the formula.

Why the Exponents Are Odd Numbers

The exponents 1.852, 0.63, and 0.54 come directly from curve-fitting experimental data. They have no physical derivation. They produce a good fit to water flow data in the range the formula was calibrated for, and significant errors outside that range. This is the fundamental limitation of all empirical formulas.

Hazen-Williams C Coefficient Table

The C value captures both pipe material roughness and pipe condition. Unlike the Darcy-Weisbach roughness ε, C also implicitly accounts for the effect of Reynolds number — which is why it should only be used for water at near-ambient temperatures, where Reynolds number effects fall within the range the formula was calibrated for.

Source: AWWA Manual M11 (Steel Pipe) and general hydraulic engineering references. C values for aged pipe are indicative only. Actual values depend heavily on water chemistry, temperature, and maintenance history.

How C Changes with Age

The degradation of C over time is highly variable and is the biggest source of uncertainty in Hazen-Williams calculations for existing systems. A new cast iron main with C = 130 can drop to C = 65 after 30 years in corrosive water conditions — cutting pipe flow capacity by more than 35% at the same head loss. Water utilities track C values through field measurement and hydraulic modelling calibration, updating their network models as mains age.

This is directly analogous to the effective roughness increase in the Darcy-Weisbach roughness framework. Both methods capture the same physical phenomenon — increased flow resistance with age — but through different parameters.

Where Each Method Applies

Use Darcy-Weisbach For:

Any fluid other than water. Oil, gas, air, steam, refrigerants, chemical process fluids — Darcy-Weisbach handles all of them as long as you have correct density and viscosity data. High-temperature or low-temperature water systems, where viscosity departs significantly from ambient conditions (hot water heating, chilled water, cryogenic). Very high or very low velocities, where Hazen-Williams errors become large. Small-diameter pipes below 50 mm where Hazen-Williams was not calibrated. Any system where accuracy matters and you have access to a calculator or spreadsheet. New designs where you want to understand sensitivity to pipe material and age. Code-compliant calculations in most mechanical and chemical engineering standards.

Use Hazen-Williams For:

Cold water distribution systems at near-ambient temperatures where historical C values are well established for the specific pipe inventory. Fire protection system hydraulic calculations, where NFPA 13 explicitly permits Hazen-Williams with tabulated C values. Municipal water supply network modelling in EPANET and similar software where the legacy database is built around C values. Situations where you are working with an existing Hazen-Williams-based dataset and converting to Darcy-Weisbach would require re-establishing all roughness values from scratch. Quick estimates where speed matters more than the last 5% of accuracy.

Where Hazen-Williams Fails

Hazen-Williams assumes the flow is turbulent, the fluid is water, and the temperature is near 15°C. Deviate from any of these and the formula produces errors it cannot warn you about, because it has no Reynolds number term to indicate when conditions are outside its calibrated range.

The most common failure mode is applying Hazen-Williams to hot water systems. A domestic hot water recirculation pipe at 60°C has water viscosity about 47% lower than at 15°C. That changes Reynolds number, flow regime, and friction factor — none of which Hazen-Williams accounts for. Using Hazen-Williams for that system produces pressure drop errors of 10 to 20%.

Accuracy and Error Analysis

The accuracy of both methods depends on how well you know the input parameters, not just which equation you use.

Darcy-Weisbach Accuracy

The Colebrook-White equation is accurate to within 1 to 2% of experimental friction factor data across its valid range. The dominant error source is usually roughness uncertainty. New pipe roughness values from manufacturer data are accurate to maybe 20%. Aged pipe roughness can vary by a factor of 3 to 10 from the new-pipe value. A 10% uncertainty in roughness produces roughly a 3 to 5% uncertainty in friction factor and therefore pressure drop. This is the irreducible error floor for Darcy-Weisbach in practice, not the equation itself.

Hazen-Williams Accuracy

Within its intended range (water, 5°C to 25°C, velocities 0.3 to 3 m/s, diameters 50 mm to 1,800 mm), Hazen-Williams is accurate to within about 10 to 15% compared to Darcy-Weisbach results. Outside this range, errors grow quickly. The table below shows typical Hazen-Williams errors at different conditions relative to Darcy-Weisbach.

The C Value vs ε Problem

A deeper accuracy issue with Hazen-Williams is that C conflates pipe roughness and Reynolds number effects into a single coefficient. In Darcy-Weisbach, roughness (ε/D) and Reynolds number are independent inputs. You can change flow velocity and see exactly how friction factor changes. In Hazen-Williams, if you change velocity, you should theoretically use a different C value — but in practice, engineers use a single C for all velocities, introducing a systematic error that is invisible from within the formula.

Worked Example: Both Methods Side by Side

A 200 mm diameter cement-lined ductile iron water main, 500 m long, carries water at 20°C at a flow rate of 50 L/s. Calculate head loss using both methods.

The large difference here comes from the C value. A cement-lined ductile iron pipe at C = 140 is treated as extremely smooth by Hazen-Williams. But Darcy-Weisbach with the correct roughness and Reynolds number produces a friction factor that reflects the actual turbulent flow conditions. This example highlights a real problem: if you use an over-optimistic C value for a new pipe, Hazen-Williams underestimates head loss and produces an undersized pump.

Recalculating with C = 120 (a more conservative new-pipe value) gives h_f = 3.11 m — still 43% below the Darcy-Weisbach result. The discrepancy reduces but does not disappear. For this flow rate and pipe, the two methods are simply not calibrated to agree, which is why Darcy-Weisbach is the more reliable choice for new designs.

Temperature Effects: Where Hazen-Williams Breaks Down

Temperature is the clearest demonstration of Hazen-Williams' limitation. As water temperature rises, viscosity falls, Reynolds number increases, and friction factor changes. Darcy-Weisbach captures this automatically if you use correct viscosity data. Hazen-Williams has no viscosity term and cannot respond to temperature at all.

Hazen-Williams gives 2.20 m at every temperature. Darcy-Weisbach correctly shows that head loss changes with temperature (viscosity reduces friction factor at higher temperatures, slightly reducing head loss). The Hazen-Williams result is not calibrated to this pipe at this flow rate — so it is consistently wrong by 58 to 60%, regardless of temperature. The error does not change with temperature because Hazen-Williams has no mechanism to respond to viscosity changes at all.

This example uses the same C = 140 value as the previous calculation. If you used a C value calibrated specifically to produce the correct answer at 20°C for this pipe (which would give approximately C = 190, outside the standard range), Hazen-Williams would be correct at 20°C but still wrong at other temperatures.

Which to Use for Your Project

A Note on Fire Suppression Design

NFPA 13 (Standard for the Installation of Sprinkler Systems) explicitly specifies Hazen-Williams with tabulated C values for sprinkler pipe materials. If you are designing a fire suppression system in the United States, you use Hazen-Williams because the standard requires it, not because it is more accurate. The C values in NFPA 13 are calibrated to produce conservative (higher) pressure drop estimates, which is appropriate for a life-safety application. Switching to Darcy-Weisbach for an NFPA 13 design requires demonstrating equivalence, which is a significant additional burden.

Converting Between the Two Methods

Sometimes you need to work in both frameworks, for example when calibrating a Darcy-Weisbach model against field data recorded as C values, or when checking a Hazen-Williams design against a Darcy-Weisbach standard.

Equivalent C Value from Darcy-Weisbach

You can derive an equivalent C value for a given set of Darcy-Weisbach conditions by equating both head loss formulas for the same Q, D, and L, then solving for C. The result depends on flow rate and pipe diameter, confirming that no single C value can represent a pipe across all conditions. This is the fundamental incompatibility between the two methods.

For a specific design point (fixed Q, D, and T), the equivalent C is:

This is complex enough that it is easier to run both calculations in parallel and compare head loss directly, rather than convert between coefficients. Use the Moody Chart Calculator for the Darcy-Weisbach side and apply Hazen-Williams separately.

Typical Equivalent C Values for Common Pipe Materials

As a rough guide, the following C values produce approximately equivalent results to Darcy-Weisbach at typical water main conditions (water at 15°C, velocity 1 to 2 m/s, diameters 100 to 400 mm):

These equivalences hold only at the specific conditions stated. Change velocity, diameter, or temperature and the equivalent C shifts. This is exactly why Hazen-Williams C and Darcy-Weisbach ε are not interchangeable parameters — they describe different physical things.

Frequently Asked Questions