Significance A new explicit formula for estimating the friction factor using six parameters is proposed. The model was set up by considering the effect of residual stresses in the flow by two distinct contributions: the first is attributed to the flow velocity (Reynolds number) and the second to the duct roughness. Compared to other models, this new equation gives the best fit with Nikuradse's results. A new model to calculate the friction is proposed. The model is based on assuming the residual stresses due to the laminar to turbulent flow transition by two distinct contributions: the first is attributed to the flow velocity (Reynolds number) and the second to the duct roughness. Compared to other models, this new equation gives the best fit with respect of Nikuradse's results. The model does not consider the effect of pipe wall on the velocity distribution. © 2019 American Institute of Chemical Engineers AIChE J, 65: 1144–1148, 2019

The friction factor is traditionally computed through Colebrook's equation or by using Moody's chart. However, these approaches have shown their limitations in getting accurate values for such factor. Different models to calculate the friction factor have been proposed, achieving different levels of certainty. In this paper, we revisit the distinct strategies used to estimate the friction factor and propose the use of a modified version of a model proposed by these authors. This model is based on the phenomenology of the flow as it transitions from laminar to turbulent flow conditions. Keywords Friction factor • Colebrook equation • Moody chart • Lambert W function • Artificial intelligence List of symbols ε Average projection of roughness, m ν Kinematic viscosity, m 2 /s ε/D Relative roughness f Friction factor Re Reynolds number and similar abbreviations do not use italics D Pipe diameter, m U Flow maximum velocity, m/s ū Flow mean velocity, m/s V* Shear force-velocity or friction velocity, m/s

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A popular equation that is often employed to represent the relationship between the pressure loss and the fluid flow in fluidized or packed granular beds is the Ergun equation, which is an extension of Darcy’s law. In this paper, the method of Smoothed Particle Hydrodynamics (SPH) is used to numerically study the flow field across a rectangular channel partially filled with a porous layer both at the Representative Elementary Volume (REV) scale using the Ergun equation and at the pore scale. Since the flow field can be estimated at the REV scale with a much lower cost compared to the pore scale, it is important to evaluate how accurately the pore-scale results can be reproduced at the REV scale. The comparison between both scales is made in terms of the velocity profiles at the outlet of the rectangular channel and the pressure losses across the clear and porous zones for three different arrays of solid grains at the pore scale. The results show that minimum differences in the flow structure and velocity profiles between the REV and the pore scale always occur at intermediate values of the porosity (ϕ=0.44 and 0.55). As the porosity increases, the differences between the REV and the pore scale also increase. The details of the pressure losses are affected by the geometry of the porous medium. In particular, we find that the pressure profiles at the REV scale match those at the pore scale almost independently of the porosity only when the grains are uniformly distributed in a non-staggered square array.