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Moody diagram friction factor

The Moody diagram, presented by Lewis F. Moody in 1944, graphically correlates the Darcy friction factor (fD) with the Reynolds number (Re) and relative roughness (ε/D), based on more than 10 000 experiments in circular pipes. It is the most widely used tool for predicting frictional head loss in fully developed incompressible flow.

Darcy‑Weisbach Equation

Section titled “Darcy‑Weisbach Equation”

The head loss hf in a straight pipe of length L and internal diameter D is proportional to the square of the mean velocity V, and is expressed by the fundamental equation:

hf = fD · (L / D) · (V² / 2g)

where g = 9.81 m/s² (32.174 ft/s²) is the acceleration due to gravity and fD is the Darcy friction factor. The associated pressure drop is:

∆p = fD · (ρ V² / 2) · (L / D) = ρ g hf

Symbol Quantity SI Unit US Unit
hf Head loss m ft
fD Darcy friction factor dimensionless dimensionless
L Pipe length m ft
D Pipe internal diameter m ft – in
V Mean fluid velocity m/s ft/s
g Acceleration due to gravity 9.81 m/s² 32.174 ft/s²
ρ Fluid density kg/m³ lb/ft³
∆p Pressure drop Pa psi – lb/in²

Darcy friction factor (fD)

Section titled “Darcy friction factor (fD)”

For laminar flow (Re ≤ 2300) the friction factor is independent of roughness and is exactly:

fD = 64 / Re

In turbulent flow (Re ≥ 4000) the dependence is more complex and is described by the implicit Colebrook‑White equation, which covers both smooth and rough pipes:

1 / √fD = –2.0 log₁₀ [ (ε/D) / 3.7 + 2.51 / (Re √fD) ]

This formula converges quickly by iteration and is the foundation of the Moody diagram. For smooth pipes and Re between 3 000 and 100 000, the Blasius approximation can be used:

fD = 0.0791 Re⁻¹/⁴

At very high Reynolds numbers (Re > 10⁵) greater accuracy is obtained with:

fD = 0.046 Re⁻¹/⁵

Moody Diagram

Section titled “Moody Diagram”

The Moody diagram plots the friction factor fD as a function of the Reynolds number Re, for relative roughness ε/D ranging from 0.000001 to 0.05. It is organized into three main zones:

  • Laminar zone (Re ≤ 2300): straight line fD = 64/Re, unique and independent of roughness.
  • Critical or transition zone (2300 < Re < 4000): the flow oscillates between laminar and turbulent; the friction factor is not defined with certainty and is usually interpolated between the laminar value at Re = 2300 and the turbulent value at Re = 4000.
  • Turbulent zone (Re ≥ 4000): family of curves that, for each ε/D, descend from the smooth pipe curve until reaching a horizontal asymptote in the fully rough regime, where fD depends only on relative roughness.

Below are typical friction factor values for smooth pipe in turbulent flow, calculated with the Colebrook‑White formula (ε = 0).

Reynolds number, Re Friction factor, fD
4 000 0.0399
10⁴ 0.0309
10⁵ 0.0180
10⁶ 0.0116
10⁷ 0.0081
10⁸ 0.0059

Absolute roughness of common materials

Section titled “Absolute roughness of common materials”

The absolute roughness of commercial steel is approximately 0.046 mm (0.0018 in), while that of cast iron reaches 0.26 mm (0.010 in). The following table compiles representative values measured in new pipes; with aging and corrosion, roughness can increase considerably.

Material Absolute roughness ε
Drawn tubing (copper, brass, glass) 0.0015 mm / 0.00006 in
Commercial steel or wrought iron 0.046 mm / 0.0018 in
Asphalted cast iron 0.12 mm / 0.0047 in
Galvanized iron 0.15 mm / 0.0059 in
Uncoated cast iron 0.26 mm / 0.010 in
Concrete 0.3 – 3 mm / 0.012 – 0.118 in
Riveted steel 0.9 – 9 mm / 0.035 – 0.354 in

Selection criteria for the friction factor

Section titled “Selection criteria for the friction factor”

For Re ≤ 2300, use directly fD = 64/Re. Between 2300 and 4000 the flow is transitional and the friction factor is uncertain; it is recommended to take the turbulent value corresponding to Re = 4000 for conservative calculations. Above 4000 the operating procedure is:

  1. Calculate the Reynolds number: Re = ρ V D / μ.
  2. Determine the relative roughness: ε/D, using the ε value from the previous table.
  3. With Re and ε/D, read fD directly from the Moody diagram, or solve the Colebrook‑White equation by iteration (Newton‑Raphson method, for example).
  4. Substitute fD into the Darcy‑Weisbach equation to obtain the head loss.

It should be noted that there is an uncertainty of at least ±10% in the fD values obtained from the diagram, due to experimental scatter and actual variations in roughness.

Frequently Asked Questions (FAQ)

Section titled “Frequently Asked Questions (FAQ)”

What is the difference between the Darcy friction factor and the Fanning friction factor?

Section titled “What is the difference between the Darcy friction factor and the Fanning friction factor?”

The Fanning factor fF is equal to one quarter of the Darcy factor (fD = 4 fF). The Moody diagram uses the Darcy factor, which is the one appearing in the Darcy‑Weisbach equation for direct calculation of head loss.

Why does the Moody diagram not cover Reynolds numbers between 2300 and 4000?

Section titled “Why does the Moody diagram not cover Reynolds numbers between 2300 and 4000?”

In that range the flow can be intermittently laminar or turbulent, depending on external disturbances, and there is no single curve. In practice, interpolation is used or the turbulent value is taken to err on the side of safety.

What is the roughness of a PVC pipe?

Section titled “What is the roughness of a PVC pipe?”

New PVC has a typical absolute roughness of 0.0015 mm (0.00006 in), similar to glass or drawn tubing. Over time, deposits may adhere and increase this value.

Is the diagram valid for any fluid?

Section titled “Is the diagram valid for any fluid?”

Yes, as long as the flow is single-phase, incompressible, and fully developed. The influence of fluid properties is captured through the Reynolds number; the diagram is dimensionless and universal.

How is the procedure when the pressure drop is known and the flow rate is unknown?

Section titled “How is the procedure when the pressure drop is known and the flow rate is unknown?”

Since fD is a function of Re, which in turn depends on flow rate, the problem is iterative. An initial value of fD is assumed (for example 0.02), velocity is calculated, Re is recalculated, and fD is corrected until convergence.

What accuracy does the Colebrook‑White equation have?

Section titled “What accuracy does the Colebrook‑White equation have?”

For commercial pipes in the range Re > 4000, the Colebrook‑White equation reproduces experimental data with an error less than 5%, provided the roughness is correctly characterized.

Sources Consulted

Section titled “Sources Consulted”