A New One‐Equation Model of Fluid Drag for Irregularly Shaped Particles Valid Over a Wide Range of Reynolds Number

Abstract

A new drag law for irregularly shaped particles is presented here. Particles are described by a shape factor that takes into account both sphericity and circularity, which can be measured via the most commonly used image particle analysis techniques. By means of the correlation of the drag coefficient versus the particle Reynolds number and the shape factor, a new drag formula, which is valid over a wide range of Reynolds number (0.03–10,000), is obtained. The new model is able to reproduce the drag coefficient of particles measured in terminal velocity experiments with a smaller scatter compared to other laws commonly used in multiphase flow engineering and volcanology. Furthermore, the new formula uses only one equation, whereas previous models, to insure validity over an ample range, made use of a step function that introduces a discontinuity at the switch of equations. Finally, this drag law works in the whole range of variation, from extremely irregular particles to perfect sphere. A code of the iterative algorithm for the drag coefficient calculation and terminal velocity is included in the supporting information both as a Fortran routine and a Matlab function.

Reply to Comment by G. Bagheri and C. Bonadonna on “A New One‐Equation Model of Fluid Drag for Irregularly Shaped Particles Valid Over a Wide Range of Reynolds Number”

“…The comment's authors challenged the definition of Re used in our paper Dioguardi et al (): where ρ f is the fluid density, w t is the particle terminal velocity, d is the particle size, and μ is the fluid viscosity. We agree with the comment's authors that (1) applies only when particles are at their terminal velocity, but this is exactly the case of the dataset published in Dioguardi et al (). In fact, our original data set includes only terminal velocity measurements in a static fluid, from which the measured drag C d was obtained by inverting the Newton's impact law: …”

“…As already pointed out in the comment 2019JB017697, our model has slightly larger average deviations than Bagheri and Bonadonna, but, regardless of the methodology for recalculating C d , it remains the model that best reproduce the measured terminal velocities, that is, with a slope of the correlation line in the scatter plot w t, calc versus w t, meas equal to 0.993 with the iterative methodology and 0.997 with the original direct methodology, which are the values of the slope closest to 1 among all models. Additionally, Dioguardi et al () model has a significantly better performance at Re < 0.1 compared to all the other considered models.…”

“…In our opinion, the assessment of the performance of the models should also include, and mainly refer to, the slope of the correlation line w t, calc versus w t, meas , since this parameter quantifies the tendency of the model to underestimate or overestimate or correctly predict w t in the whole investigated experimental range. Results of the intercomparison study considering all the drag models originally taken into account (Bagheri & Bonadonna, ; Bagheri & Bonadonna, ; Chien, ; Dellino et al, ; Dioguardi et al, ; Dioguardi & Mele, ; Ganser, ; Haider & Levenspiel, ; Pfeiffer et al, ; Swamee & Ojha, ) can be found in the supporting information file “dioguardi_ds03.xlsx”; for brevity, here we focus on the models that are more often used or referred to in volcanology. Table summarizes the average error and the RMSE calculated using equations and , respectively: where N is the number of experimental data points.…”

“…All results refer both to our original recalculation methodology and the iterative one. The performance of our model (Dioguardi et al, ) with both the original and the iterative methodology are also presented in Figure . As already pointed out in the comment 2019JB017697, our model has slightly larger average deviations than Bagheri and Bonadonna, but, regardless of the methodology for recalculating C d , it remains the model that best reproduce the measured terminal velocities, that is, with a slope of the correlation line in the scatter plot w t, calc versus w t, meas equal to 0.993 with the iterative methodology and 0.997 with the original direct methodology, which are the values of the slope closest to 1 among all models.…”

“…(a) Calculated versus measured C d versus Re with Dioguardi et al () model with the original direct methodology. (b) Calculated versus measured w t and correlation line obtained with Dioguardi et al () with the original direct methodology. (c) Calculated versus measured C d versus Re with Dioguardi et al () model with the iterative methodology.…”