The Moody Diagram¶

The Moody diagram relates the Darcy friction factor, \(f\), to Reynolds number and relative surface roughness. Pyskyfire uses this friction factor when it calculates pressure loss in a cooling channel.

Quantities in the diagram¶

The Reynolds number based on hydraulic diameter is

\[ Re_{D_h} = \frac{\rho u D_h}{\mu}, \]

where \(\rho\) is the fluid density, \(u\) is the mean channel velocity, \(D_h\) is the hydraulic diameter, and \(\mu\) is the dynamic viscosity. Each curve in the diagram represents a different relative roughness, \(\epsilon/D_h\), where \(\epsilon\) is the absolute surface roughness.

For fully developed flow through a channel of length \(L\), the friction factor appears in the Darcy–Weisbach pressure-loss equation:

\[ \Delta p = f\frac{L}{D_h}\frac{\rho u^2}{2}. \]

Friction-factor model¶

Pyskyfire treats flow below \(Re_{D_h}=2300\) as laminar and uses

\[ f_{lam} = \frac{64}{Re_{D_h}}. \]

For turbulent flow with a specified roughness, it solves the Colebrook–White equation iteratively:

\[ \frac{1}{\sqrt{f_{turb}}} + 2\log_{10}\left( \frac{\epsilon}{3.71D_h} + \frac{2.51}{Re_{D_h}\sqrt{f_{turb}}} \right) = 0. \]

When no roughness is specified, the smooth-wall turbulent expression is

\[ f_{turb} = \left(0.79\ln Re_{D_h} - 1.64\right)^{-2}. \]

Between \(Re_{D_h}=2300\) and \(Re_{D_h}=3500\), Pyskyfire uses a linear blend rather than an abrupt switch. Defining

\[ \alpha = \frac{Re_{D_h}-2300}{3500-2300}, \]

the blended friction factor is

\[ f = (1-\alpha)f_{lam} + \alpha f_{turb}. \]

Above \(Re_{D_h}=3500\), the turbulent result is used directly.

How the chart is generated¶

tools/generate_engineering_charts.py creates the diagram with pyskyfire.viz.PlotMoodyDiagram. The plot evaluates the same pyskyfire.regen.f_darcy function used by the cooling solver at 400 logarithmically spaced Reynolds numbers from \(7\times10^2\) to \(10^8\). It sets \(D_h=1\) so the absolute roughness value passed to the calculation is also the relative roughness \(\epsilon/D_h\). Each roughness curve is therefore a direct visualisation of the implemented friction-factor model.