穆迪圖

概述

The Moody diagram is a dimensionless chart relating the Darcy friction factor (λ or f) to Reynolds number (Re) and relative roughness (k/d) for fully developed flow in circular pipes. It is the standard tool for estimating pipe friction losses in the Darcy-Weisbach equation and covers laminar, transitional, and turbulent flow regimes.

Two conventions exist for the friction factor. The SI (Darcy) friction factor λ is four times the Imperial (Fanning) friction factor f. Always confirm which convention applies before using a value.

關键公式

Reynolds Number

$\text{Re} = \frac{\rho \, v \, d}{\mu} = \frac{v \, d}{\nu}$

Determines whether flow is laminar (Re < 2300), transitional (2300 < Re < 4000), or turbulent (Re > 4000).

Darcy-Weisbach Head Loss

$h_f = \lambda \, \frac{L}{d} \, \frac{v^2}{2g}$

Major (friction) head loss over pipe length L.

Relative Roughness

$r = \frac{k}{d}$

Ratio of pipe absolute roughness to inner diameter; this locates the correct curve on the Moody diagram.

Colebrook-White Equation (implicit, turbulent)

$\frac{1}{\sqrt{\lambda}} = -2.0 \, \log_{10}\!\left(\frac{k/d}{3.7} + \frac{2.51}{\text{Re}\,\sqrt{\lambda}}\right)$

An implicit correlation that reproduces the turbulent zone of the Moody diagram. Must be solved iteratively.

Laminar Friction Factor

$\lambda = \frac{64}{\text{Re}}$

Exact for all laminar flow (Re < 2300).

變量

Convention relationship: λ = 4f

Absolute Roughness for Common Pipe Materials

Roughness by Flow Regime

Relative Roughness Calculator

Head Loss Calculator

單位換算器

Interactive Moody Diagram Data

The original Moody diagram image is represented below as calculated friction-factor curves. The laminar curve uses λ = 64/Re; turbulent curves use the Swamee-Jain explicit approximation for selected relative roughness values.

Example - Reading the Moody Diagram

For a PVC pipe with very small relative roughness and Reynolds number near (10^7), the Moody diagram is read on the turbulent smooth-pipe side. The Darcy friction factor is approximately ( \lambda \approx 0.008 - 0.01 ), depending on the exact roughness assumption. This value can then be used directly in the Darcy-Weisbach equation for SI calculations. If a Fanning friction factor is required instead, divide the Darcy value by four.

Laminar vs Turbulent Friction Comparison

原始源圖

The following original source images are preserved to avoid losing visual reference material. When an image contains chart or tabular data, its extracted values are represented in the page tables, calculators, or interactive charts; remaining images are retained as visual source references.

Moody diagram

工程要點

• Transient zone caution (2300 < Re < 4000): The friction factor cannot be reliably determined because flow alternates between laminar and turbulent states. Conservative practice is to use the higher (turbulent) friction factor value at Re = 4000 for design.
• Convention trap: The Darcy (λ) and Fanning (f) friction factors differ by a factor of four (λ = 4f). Head loss equations published with f require a factor of four more than those with λ. Always confirm which version is in use.
• Roughness changes over time: New-pipe roughness values assume clean surfaces. Corrosion, scale, and biological growth can increase effective roughness by an order of magnitude in service. Apply safety factors accordingly.
• Colebrook-White is implicit: Because λ appears on both sides of the Colebrook-White equation, it requires iteration. The Swamee-Jain approximation provides an explicit alternative accurate to about ±1% for 10⁻⁶ < k/d < 10⁻² and 5000 < Re < 10⁸.
• Non-circular pipes: Use hydraulic diameter dₕ = 4A/P in place of d. The Moody diagram then provides an approximation, but accuracy decreases as the cross-section deviates from circular.
• Minor vs major losses: The Moody diagram addresses only major (friction) losses. Fittings, bends, and valves are accounted for separately as minor losses using loss coefficients K or equivalent lengths.

參考资料

• Engineering ToolBox — Moody Diagram
• Colebrook, C. F. "Turbulent Flow in Pipes." Journal of the Institution of Civil Engineers, 1939.
• Moody, L. F. "Friction Factors for Pipe Flow." Transactions of the ASME, 1944.