The Moody diagram gives the Darcy friction factor, f, in...

The Moody diagram gives the Darcy friction factor, $f,$ in terms of Reynolds number and relative roughness. The Fanning friction factor for pipe flow is defined as $$f_{F}=\frac{\tau_{w}}{\frac{1}{2} \rho \nabla^{2}}$$ where $\tau_{w}$ is the wall shear stress in the pipe. Show that the relation between the Darcy and Fanning friction factors for fully developed pipe flow is given by $f=4 f_{F}$

Key Concepts

Fully Developed Pipe Flow

Fully developed pipe flow refers to a condition where the velocity profile of the fluid does not change along the length of the pipe. This assumption simplifies the analysis by ensuring a constant shear stress distribution along the wall, thereby allowing for a clear and direct relationship between friction factors and flow parameters.

Darcy Friction Factor

The Darcy friction factor is a dimensionless quantity used to characterize the pressure losses due to friction in pipe flow. It is typically used in engineering calculations for head loss and pressure drop in pipelines, and it is defined based on the relationship between the wall shear stress and the mean flow velocity over a pipe’s characteristic length.

Fanning Friction Factor

The Fanning friction factor is another dimensionless measure of frictional resistance in a flow system, defined using the wall shear stress and the square of the fluid velocity. It is commonly used in both laminar and turbulent flow analyses and, although it is formulated differently, it is directly related to the Darcy friction factor by a constant scaling factor.

Relationship Between Friction Factors

The demonstration that the Darcy friction factor is four times the Fanning friction factor (f = 4 f_F) arises from the differences in their definitions. This relationship is crucial for converting between different formulations and ensuring consistency in the calculation of pressure drops in fully developed pipe flows.

Reynolds Number and Relative Roughness

Although not directly used to derive the 4:1 relationship between the Darcy and Fanning friction factors, the Reynolds number (which quantifies the flow regime) and the relative roughness (which influences the wall friction effects) are integral concepts in the overall analysis of pipe flow, as illustrated by the Moody diagram. These parameters help determine the appropriate friction factor for different flow conditions.

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