A Note on the Moody Diagram

In this work, we underscore the significance of selecting an appropriate scaling to derive dimensionless quantities that accurately reflect their dimensional counterparts, thereby enhancing the comprehension of the underlying physics. For the loss of head in a pipe flow, we argue that employing inertial force (or kinetic energy) to non-dimensionalized pressure force (or mechanical energy loss) lacks physical justification. As a result, an anomalous trend emerges for the classical friction factor: it decreases as the dimensionless flow rate (Reynolds number) increases, contrary to the behavior observed in the corresponding dimensional quantities. Conversely, by non-dimensionalizing the pressure force with the viscous force, a novel friction factor arises. In laminar flow, it is constant, while in turbulent flow, it is a monotonically increasing function of the Reynolds number, mirroring the behavior observed in the dimensional problem.