Moody chart

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.

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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 ${\displaystyle f_{D}}$):

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

Because this equation is implicit in ${\displaystyle f_{D}}$, solving it requires either the chart or an iterative numerical method; online calculators can also solve it directly for a given Re and ${\displaystyle \epsilon /D}$.^[7]

To read a friction factor from the chart, first compute the Reynolds number Re and the relative roughness ${\displaystyle \epsilon /D}$ of the pipe. Locate Re on the horizontal axis, trace a vertical line to the curve for that ${\displaystyle \epsilon /D}$ (interpolating between plotted curves if needed), and read ${\displaystyle f_{D}}$ from the vertical axis at that point. Below Re ≈ 2,300, a single line applies regardless of roughness, since viscous forces rather than surface irregularities govern laminar flow. Between roughly Re = 2,300 and 4,000, flow is unstable and the chart gives no reliable value.

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Solving the Colebrook equation by iteration is often impractical, so several explicit approximations have been proposed. The Swamee–Jain equation^[8] gives:

${\displaystyle f_{D}={\frac {0.25}{\left[\log _{10}\left({\frac {\epsilon /D}{3.7}}+{\frac {5.74}{\mathrm {Re} ^{0.9}}}\right)\right]^{2}}}}$

The Haaland equation^[9] offers a similar approximation:

${\displaystyle {\frac {1}{\sqrt {f_{D}}}}=-1.8\log _{10}\left[\left({\frac {\epsilon /D}{3.7}}\right)^{1.11}+{\frac {6.9}{\mathrm {Re} }}\right]}$

and the Churchill equation^[10] spans the laminar, transitional, and turbulent regimes in a single expression, making it well suited to computer implementation. These explicit forms, along with numerical Colebrook solvers, power most modern software and online calculators that compute ${\displaystyle f_{D}}$ without the printed chart.

The Moody chart and the Colebrook equation describe steady, fully developed, single-phase flow of a Newtonian fluid in a circular pipe with uniform roughness. They exclude entrance effects and pipe fittings (handled separately via minor loss coefficients), non-circular ducts (approximated using hydraulic diameter), compressible flow at high Mach number, and non-Newtonian fluids. Because the chart is built from data on commercial pipe roughness, unusual or highly non-uniform roughness can produce friction factors that diverge from its predictions.

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:

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