The Darcy-Weisbach equation is the most accurate method for calculating head loss due to friction in a pipe. It is widely used by hydraulic engineers and designers to size pipes, select pumps, and design water distribution systems. The key parameter in this equation is the friction factor f, which depends on the Reynolds number and the relative roughness of the pipe. This article explains how to determine f using the Moody chart, empirical formulas, and iterative methods.

Understanding the Darcy-Weisbach Equation

The Darcy-Weisbach equation expresses the head loss hf (in meters of fluid) as:

hf = f · (L/D) · (V²/(2g))

where:

  • f = Darcy-Weisbach friction factor (dimensionless)
  • L = pipe length (m)
  • D = pipe inner diameter (m)
  • V = average flow velocity (m/s)
  • g = acceleration due to gravity (9.81 m/s²)

To apply this equation, the friction factor must be determined accurately. The Moody chart, developed by Lewis Ferry Moody in 1944, provides a graphical solution relating f to the Reynolds number (Re) and the relative roughness (ε/D). The chart covers laminar flow (Re < 2000), transitional flow (2000 < Re < 4000), and turbulent flow (Re > 4000).

Step 1: Determine the Reynolds Number

The Reynolds number is a dimensionless quantity that characterizes flow regime:

Re = (ρ · V · D) / μ

or equivalently Re = (V · D) / ν, where:

  • ρ = fluid density (kg/m³)
  • μ = dynamic viscosity (Pa·s)
  • ν = kinematic viscosity (m²/s)

For water at 20°C, ν ≈ 1.004 × 10⁻⁶ m²/s. For example, if water flows at 1.5 m/s through a 0.1 m diameter pipe, Re = (1.5 × 0.1) / (1.004e-6) ≈ 149,400, indicating turbulent flow.

Step 2: Determine Relative Roughness

Relative roughness is the ratio of the pipe wall roughness height ε (in mm) to the pipe inner diameter D (in mm). Typical values of ε for common pipe materials are:

  • Drawn tubing (copper, brass): 0.0015 mm
  • Commercial steel: 0.046 mm
  • Cast iron: 0.26 mm
  • Galvanized iron: 0.15 mm
  • Concrete: 0.3–3.0 mm
  • PVC, HDPE: 0.0015 mm (smooth)

For a 100 mm diameter commercial steel pipe, ε = 0.046 mm, so ε/D = 0.046/100 = 0.00046.

Step 3: Use the Moody Chart or Empirical Formulas

Laminar Flow (Re < 2000)

For laminar flow, the friction factor is independent of roughness and given by f = 64 / Re. This is derived from the Hagen–Poiseuille equation.

Turbulent Flow (Re > 4000)

For turbulent flow, the Colebrook equation is the standard implicit formula:

1/√f = -2 log₁₀[ (ε/D)/3.7 + 2.51/(Re√f) ]

This equation requires iteration. Many engineers use the Swamee-Jain explicit approximation, valid for 10⁻⁶ ≤ ε/D ≤ 10⁻² and 5000 ≤ Re ≤ 10⁸:

f = 0.25 / [ log₁₀( ε/(3.7D) + 5.74/Re⁰·⁹ ) ]²

For example, with Re = 149,400 and ε/D = 0.00046, the Swamee-Jain formula gives f ≈ 0.019.

Transitional Flow (2000 < Re < 4000)

In the transitional region, flow is unstable and the friction factor is uncertain. The Moody chart shows a shaded band. For design, it is conservative to use the turbulent value or avoid operating in this range.

Step 4: Iterative Solution of Colebrook Equation

When using the Colebrook equation, an iterative method such as the Newton-Raphson method can be employed. Start with an initial guess (e.g., f₀ = 0.02) and update using:

fnew = 1 / [ -2 log₁₀( (ε/D)/3.7 + 2.51/(Re√f₀) ) ]²

Repeat until convergence (change < 0.1%). Many calculators and spreadsheet tools automate this. For manual use, the Moody chart is still common in textbooks.

Practical Example: Sizing a Pump for a Water Line

Consider a 200 m long galvanized iron pipe (ε = 0.15 mm) with inner diameter 50 mm carrying water at 20°C at a flow rate of 0.01 m³/s. Calculate the friction factor and head loss.

  1. Compute velocity: V = Q/A = 0.01 / (π × 0.025²) ≈ 5.09 m/s
  2. Compute Re: Re = (5.09 × 0.05) / 1.004e-6 ≈ 253,500 (turbulent)
  3. Relative roughness: ε/D = 0.15 / 50 = 0.003
  4. Using Swamee-Jain: f = 0.25 / [log₁₀(0.003/3.7 + 5.74/253500⁰·⁹)]² ≈ 0.026
  5. Head loss: hf = 0.026 × (200/0.05) × (5.09²/(2×9.81)) ≈ 68.7 m

This head loss helps select a pump with sufficient head rating. For more complex systems, see our complete guide to hydraulic calculations.

Comparison with Hazen-Williams Formula

The Hazen-Williams formula is simpler but less accurate for fluids other than water and for pipes with high roughness or low temperatures. It uses a coefficient C (e.g., 140 for new PVC, 100 for cast iron). For the example above, Hazen-Williams with C=100 would yield a different head loss. Learn more in our article Hazen-Williams vs Darcy-Weisbach.

Online Tools and Spreadsheets

Many free online calculators implement these formulas. For instance, the Engineering Toolbox and Pipe Flow Expert provide friction factor calculators. Spreadsheet software like Microsoft Excel can solve the Colebrook equation using goal seek. A typical Excel formula for f uses circular references, but simpler explicit formulas are often preferred.

For a comprehensive set of hydraulic calculation tools, visit our Hydraulic Calculator resource page.

Related Articles

  • The Complete Guide to Hydraulic Calculations for Engineers and Designers
  • Hazen-Williams vs Darcy-Weisbach
  • Pipe Flow Velocity Calculations
  • Head Loss in Pipe Fittings
  • Selecting the Right Pipe Material