# Moody chart

In engineering, the Moody chart or Moody diagram (also Stanton diagram) is a graph in non-dimensional form that relates the Darcy–Weisbach friction factor fD, Reynolds number Re, and surface roughness for fully developed flow in a circular pipe. It can be used to predict pressure drop or flow rate down such a pipe.

Moody diagram showing the Darcy–Weisbach friction factor fD plotted against Reynolds number Re for various relative roughness ε / D

## History

In 1944, Lewis Ferry Moody plotted the Darcy–Weisbach friction factor against Reynolds number Re for various values of relative roughness ε / D.[1] This chart became commonly known as the Moody chart or Moody diagram. It adapts the work of Hunter Rouse[2] but uses the more practical choice of coordinates employed by R. J. S. Pigott,[3] whose work was based upon an analysis of some 10,000 experiments from various sources.[4] Measurements of fluid flow in artificially roughened pipes by J. Nikuradse[5] were at the time too recent to include in Pigott's chart.

The chart's purpose was to provide a graphical representation of the function of C. F. Colebrook in collaboration with C. M. White,[6] which provided a practical form of transition curve to bridge the transition zone between smooth and rough pipes, the region of incomplete turbulence.

## Description

Moody's team used the available data (including that of Nikuradse) to show that fluid flow in rough pipes could be described by four dimensionless quantities: Reynolds number, pressure loss coefficient, diameter ratio of the pipe and the relative roughness of the pipe. They then produced a single plot which showed that all of these collapsed onto a series of lines, now known as the Moody chart. This dimensionless chart is used to work out pressure drop, ${\displaystyle \Delta p}$ (Pa) (or head loss, ${\displaystyle h_{f}}$(m)) and flow rate through pipes. Head loss can be calculated using the Darcy–Weisbach equation in which the Darcy friction factor ${\displaystyle f_{D}}$ appears :

${\displaystyle h_{f}=f_{D}{\frac {L}{D}}{\frac {V^{2}}{2\,g}};}$

Pressure drop can then be evaluated as:

${\displaystyle \Delta p=\rho \,g\,h_{f}}$

or directly from

${\displaystyle \Delta p=f_{D}{\frac {\rho V^{2}}{2}}{\frac {L}{D}},}$

where ${\displaystyle \rho }$ is the density of the fluid, ${\displaystyle V}$ is the average velocity in the pipe, ${\displaystyle f_{D}}$ is the friction factor from the Moody chart, ${\displaystyle L}$ is the length of the pipe and ${\displaystyle D}$ is the pipe diameter.

The chart plots Darcy–Weisbach friction factor ${\displaystyle f_{D}}$ against Reynolds number Re for a variety of relative roughnesses, the ratio of the mean height of roughness of the pipe to the pipe diameter or ${\displaystyle \epsilon /D}$.

The Moody chart can be divided into two regimes of flow: laminar and turbulent. For the laminar flow regime (${\displaystyle Re}$< ~3000), roughness has no discernible effect, and the Darcy–Weisbach friction factor ${\displaystyle f_{D}}$was determined analytically by Poiseuille:

${\displaystyle f_{D}=64/\mathrm {Re} ,{\text{for laminar flow}}.}$

For the turbulent flow regime, the relationship between the friction factor ${\displaystyle f_{D}}$ the Reynolds number Re, and the relative roughness ${\displaystyle \epsilon /D}$is more complex. One model for this relationship is the Colebrook equation (which is an implicit equation in ${\displaystyle f_{D}}$):

${\displaystyle {1 \over {\sqrt {f_{D}}}}=-2.0\log _{10}\left({\frac {\epsilon /D}{3.7}}+{\frac {2.51}{\mathrm {Re} {\sqrt {f_{D}}}}}\right),{\text{for turbulent flow}}.}$

## Fanning friction factor

This formula must not be confused with the Fanning equation, using the Fanning friction factor ${\displaystyle f}$, equal to one fourth the Darcy-Weisbach friction factor ${\displaystyle f_{D}}$. Here the pressure drop is:

${\displaystyle \Delta p={\frac {\rho V^{2}}{2}}{\frac {4fL}{D}},}$

## References

1. ^ Moody, L. F. (1944), "Friction factors for pipe flow" (PDF), Transactions of the ASME, 66 (8): 671–684, archived (PDF) from the original on 2019-11-26
2. ^ Rouse, H. (1943). Evaluation of Boundary Roughness. Proceedings Second Hydraulic Conference, University of Iowa Bulletin 27.
3. ^ Pigott, R. J. S. (1933). "The Flow of Fluids in Closed Conduits". Mechanical Engineering. 55: 497–501, 515.
4. ^ Kemler, E. (1933). "A Study of the Data on the Flow of Fluid in Pipes". Transactions of the ASME. 55 (Hyd-55-2): 7–32.
5. ^ Nikuradse, J. (1933). "Strömungsgesetze in Rauen Rohren". V. D. I. Forschungsheft. Berlin. 361: 1–22. These show in detail the transition region for pipes with high relative roughness (ε / D > 0.001).
6. ^ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws". Journal of the Institution of Civil Engineers. London, England. 11 (4): 133–156. doi:10.1680/ijoti.1939.13150.