A unified model for friction factor in rough pipes

What is it about?

Predicting the friction factor is essential for the design and operation of pipelines, ducts, and other fluid-transport systems. In industrial applications, turbulent flows span a broad range of Reynolds numbers and interact with surfaces exhibiting roughness features at multiple scales. These surfaces may range from hydraulically smooth walls and small sand-grain-type textures to larger, structured industrial elements, with the different roughness scales acting either independently or in combination. However, widely used correlations are typically formulated in terms of a single relative roughness parameter, (ε/D), commonly interpreted through the concept of equivalent roughness. Although this representation performs well for conventional rough surfaces, it may not fully capture the frictional behavior of pipes containing large, structured, or otherwise industrially relevant roughness-arranged elements. This work presents a unified two-parameter model for predicting the friction factor in turbulent rough-wall flows. The model separates two distinct contributions to flow resistance: the distributed effect associated with sand-grain-type roughness and the additional drag generated by larger roughness elements that extend beyond the near-wall region. This distinction recognizes that the two roughness classes produce different mechanisms of momentum loss and, consequently, cannot always be represented adequately by a single equivalent roughness parameter. The proposed formulation is explicit, eliminating the need for iterative calculations, and provides a continuous description of the friction factor over a broad range of Reynolds numbers and surface conditions. It recovers classical behaviors, including the Blasius regime for hydraulically smooth turbulent flow, while also extending to high-Reynolds-number conditions and roughness configurations relevant to industrial systems. By incorporating conventional and large-scale roughness effects within a single equation, the model provides a more comprehensive, physically informed, and flexible framework for supporting reliable pressure-drop and energy-demand predictions in more realistic engineering applications.

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Photo by Fern M. Lomibao on Unsplash

Photo by Fern M. Lomibao on Unsplash

Why is it important?

The Colebrook equation, often represented by the Moody chart, uses a single relative roughness parameter and may not capture the diverse drag mechanisms associated with complex industrial surfaces. Explicit approximations share this limitation, whereas the proposed two-parameter model distinguishes the effects of small-scale, sand-grain-type roughness from those of larger industrial roughness elements. For engineering applications, the model offers three main advantages. First, it retains the familiar structure of classical friction-factor approaches, using Reynolds number and roughness information as the main inputs, while providing a unified prediction framework from hydraulically smooth conditions to highly rough pipes. Second, its explicit form avoids iterative calculations and facilitates implementation in engineering spreadsheets, design tools, network solvers, CFD workflows, and process simulations. Third, the two-parameter formulation provides clearer insight into how different roughness scales contribute to flow resistance and pressure losses. Overall, the model supports more reliable, efficient, and scalable friction-factor predictions while complementing established approaches and extending their applicability to more complex and industrially relevant surface conditions.

Perspectives