1. Introduction
During suspensions processing, it is necessary to have information about the particle distribution. The settling of particles in the gravitational field is the most common case. The ability to determine the settling velocity of particles in suspension is necessary when designing equipment or assessing a change in suspension composition. The basic chemical engineering calculation is simple in the case of low-concentrate suspensions formed by spherical or isometric particles. In the case of concentrated fibrous suspensions, i.e., suspensions formed by significantly non-isometric or 1D particles, the determination of the settling velocity is very complicated. Another complication is if the liquid exhibits non-Newtonian behavior.
A typical example of such suspensions is fresh fine-grained cementitious composite reinforced by distributed steel fibers. The distribution of the fibers in this matter due to particle sedimentation has an effect on the mechanical properties of constructs made from these cementitious composites. The mechanical properties of this composite are well known. During manufacturing of structures, however, fiber sedimentation and non-homogeneity of the structure can occur. The areas with minimal or non-uniform fibers have a great impact on the mechanical properties of structures and, can have a critical impact on the load-bearing capacity of constructed elements. In the existing literature, it is difficult to find information about the settling velocity of non-isometric particles which are relatively long in diameter and can become interweaved and connected. The global production of varieties of concrete mixtures which exhibit pseudo-plastic behavior is in the billions of tons per year. This means that knowledge about the hindered settling of significantly non-isometric particles in non-Newtonian pseudo-plastic liquids has a huge impact on production of HPC/UHPC structures.
For these reasons, the focus of this article is on the hindered settling of suspension clouds of non-isometric particles with a longitudinal dimension several orders of magnitude larger than their transverse dimension, specifically, with ratio d/l = 0.3/20, which are commonly used in concrete structures. These steel fibers are used as filler in modern HPC/UHPC mixtures. The fresh state of fine-grained cementitious HPC/UHPC composites exhibits the non-Newtonian behavior of a pseudo-plastic liquid.
The properties of fine-grained cementitious composites (HPC and UHPC) in both fresh and hardened states depends on the mixture’s composition as well as on the type of dispersion fiber and the production technology of the concrete element (see [
1,
2,
3]). The shape, size, and material properties of the fibers have a significant impact on the rheology and workability of the fresh state of the mixture. Fibers can settle the process of concrete hardens, causing non-homogeneity in the structure. For this reason, the compaction of the mixture is reduced during production of HPC/UHPC elements to prevent fiber segregation. Non-homogenous fiber distribution or its concentration in specific areas can lead to several problems in structural elements. Areas with minimal reinforcing fibers or with non-uniform reinforcement can cause structural deformation, e.g., cracks. These areas have a critical impact on the load bearing capacity of elements impacted by the homogeneity of fiber distribution and the orientation of fibers [
4]. To predict the non-homogeneity of the mixture during processing, it is necessary to have a simple model describing the dependence of the settling speed of steel reinforcing particles on the concentration and flow properties of the liquid phase by modelling the behavior of the fresh concrete.
The settling velocity of one particle in an unlimited area, called the free settling velocity, can be determined from Equation (1), which is based on the balance of forces acting on a single particle [
5]:
where
u is the hindered settling velocity of the particle cloud,
D is the characteristic particle diameter,
ρs is the density of the particles,
ρ is the liquid density,
g is the gravitational acceleration, and
CD is the drag coefficient.
The hindered settling velocity is affected by the drag coefficient, CD, as shown in Equation (1). The drag coefficient CD of a non-spherical particle during sedimentation depends on both the shape of the particle and on its orientation. Non-spherical particles have a tendency to rotate or flip during sedimentation. For this reason, it is difficult to accurately determine the drag coefficient CD of non-spherical particles during sedimentation. A number of authors have attempted to determine the drag coefficient of non-spherical particles.
Chhabra and Richardson [
6] presented a simple relationship for the drag coefficient of cones, cubes, parallelepipeds, and axially falling cylinders in power-law fluids. Their equation is based on replacing a non-spherical particle with a given diameter with a spherical particle of the same volume. However, the authors point out that the accuracy of this relationship decreases with decreasing sphericity of the real particle. Loth [
7] dealt with the drag coefficient of non-spherical particles of various shapes (regular and irregular) and for various Reynolds particle numbers in free fall. The author found that to determine the drag coefficient it is best to replace the sphericity of the particles with the aspect ratio for spheroidal particles, the surface area ratio for common non-spherical particles, and the max-med-min area ratio for very irregular particles. These substitutions apply to both high Reynolds numbers and low Reynolds numbers. However, the author points out that even with this considered substitution, the drag coefficient is determined by position of the particle at a given location during the fall; these shape substitutions are therefore only approximate, because, e.g., regularly shaped particles of different shapes with the same sphericity ratio may have different drag coefficients. Chhabra et al. [
8] compared models proposed by different authors for determining the drag coefficient of non-spherical particles. The authors created a database of available data consisting of about 1900 datapoints. This database contains different particle shapes, their densities, and the different densities and viscosities of fluids covering a wide range of Reynolds numbers. The authors quantified the shape of non-spherical particles using sphericity (ranging from 0.1 for a disk to 1.0 for a sphere). The authors found that the best results were obtained using the Ganser method. This method uses the diameter of a spherical particle with equal volume as the characteristic dimension of the non-spherical particle. The authors further found that while the error of the compared methods can be 16, it can be up to 100% depending on the sphericity of the particle, with lower sphericity of the particle indicating lower agreement. In contrast to this work, Yow et al. [
9] used the Kaskas equation to calculate the drag coefficient of a regular particle, which is much simpler to use and sufficiently accurate. The authors complemented this equation with coefficients dependent only on sphericity. It is clear from this research that it is not possible to precisely determine the drag coefficient of a non-spherical particle on account of the many factors affecting its size, which are most variable during a fall (especially rotation of a non-spherical particle during sedimentation). In their work, the authors recommend carrying out experiments with non-spherical particles to understand their behavior while falling before beginning separate experiments.
As mentioned above, the drag coefficient is an important parameter influencing the settling velocity. In recent years, increased attention has been paid to settling velocities in both Newtonian and non-Newtonian fluids. The settling velocity of a spherical particle in a non-Newtonian fluid was discussed by Machač et al. [
10]. The authors found that low-concentration kaolin suspensions in a creep region can be considered as power-low fluids. According to the authors, much more accurate determination of the settling velocity can be obtained using the Herschel–Bulkley model. Agarwal and Chhabra [
11] published results of experiments determining the settling velocity of cubes with a sphericity of 0.805 for different power law index sizes (0.61–1), consistencies, and Reynolds numbers, and verified the validity of the drag coefficient determination.
In cases of sedimentation of larger number of particles, such as the sedimentation of fibers during the processing of filled UHPC mixtures, hindered settling occurs. Furthermore, the hindered settling also occurs near the area of solid walls. In the case of industrial applications in large plants, hindered settling near the wall can be neglected in comparison with the hindered settling caused by the interaction of particles. Then, the free settling velocity must be correlated only in the presence of a larger number of particles, expressed by their concentration in the cloud of suspended particles.
The most widely used model for calculating settling velocity during hindered settling of particles is that of Richardson and Zaki [
12], usually expressed as the dependence on the porosity of the suspension cloud in the form
where
u is the hindered settling velocity of the particle cloud,
uinf is the settling velocity of one particle in unlimited area,
ε is the porosity of the suspension cloud and is defined on the base of the volumetric particle concentration
cv by relation
ε = 1 −
cv, and
a is the Richardson–Zaki constant. The Richardson–Zaki constant
a is equal to 4.6 for hindered settling in the Stokes regime.
Kramer et al. [
13] correlated the Richardson–Zaki constant and mentioned the value of the Richardson–Zaki constant as being 4.8. Di Felice and Kehlenbeck [
14] verified this model on spherical particles of various materials. The same authors pointed out the need for correlation of the Richardson–Zaki constant at certain ratios of particle diameter and vessel diameter, as the settling velocity of particles is affected by the vessel wall. A generalized approach to modelling the settling velocity of spherical particles is provided in [
15]. The authors used a Newtonian fluid and three non-Newtonian fluids for different Reynolds numbers along with the drag coefficient to obtain the method.
Recently, a number of works have attempted to specify the Richardson–Zaki constant in a way that it is valid for non-spherical particles. Alrawi et al. [
16] determined the value of the Richardson–Zaki constant for non-spherical irregular natural sediments in a concentrated suspension. The equations of the settling velocity of spherical and non-spherical (cube, cylinder) particles in a Newtonian fluid were described in [
17]. The authors found that in order to accurately determine the drag coefficient and settling velocity of non-spherical particles in a Newtonian fluid it is important to determine the shape of the particle and its orientation during settling. Furthermore, the authors created a correlation for the drag coefficient of spherical and non-spherical particles, including the effect of particle sphericity and orientation during sedimentation. The authors proposed a settling velocity equation that directly predicts the particle settling velocity of particles with different shapes and Reynolds numbers ranging from 0.471 to 1; a suitable range of particle shapes in this model includes spheres, cubes and cylinders in the ratio
d:
l = 1:5, 1:3, 5:1, 3:1, 1:1, 10:1. Bagheri and Bonadonna [
18] investigated the influence of the shape, surface roughness, orientation, and ratio of particle density to fluid on the drag coefficient value of non-spherical particles of regular and irregular shapes in the subcritical region of the Reynolds number. They presented a general correlation of the drag coefficient under the assumption of random particle orientation in the Stokes region and Newton region. The authors found that the effect of particle orientation is significant at high values of the Reynolds number. Furthermore, they found that the drag coefficient in the Stokes region is more sensitive to changes in elongation than to changes in flatness.
4. Conclusions
In the literature, a great deal of information can be found on the free and hindered settling of isometric particles in Newtonian fluids. In addition, several articles describe the behavior of isometric particles in non-Newtonian fluids. Based on our study of the free and hindered settling of significantly non-isometric particles, we have found that it is not possible to use these data to describe the sedimentation of such particles.
For this reason, systematic experiments were carried out to describe the hindered settling of suspension clouds of non-isometric particles with a longitudinal dimension several orders of magnitude larger than their transverse dimension (d/l = 0.3/20). Similar fibers are used as filler in modern HPC/UHPC mixtures. Experiments were carried out in transparent model fluids, namely, glycerine (a Newtonian fluid) and an aqueous solution of carboxylmethylcelulose CMC (a non-Newtonian pseudo-plastic liquid), as these have similar rheological properties to fresh HPC/UHPC mixtures. A correlation was proposed to calculate the hindered settling velocity of the cloud of these fibers depending on the porosity of the suspension and the flow properties of the liquid phase of the suspension.
First, we carried out experiments describing the settling velocity of one fiber particle depending on fiber orientation during sedimentation in the Stokes regime. Our experiments showed that the settling velocity of the fiber particle was orders of magnitude smaller than the settling velocity produced on the basis of the suggested correlation with an equivalent spherical particle. A more noticeable decrease in the average value of the fiber settling velocity was observed only in case of horizontal insertion of the fiber into the liquid, when the particle fiber changed its orientation to a natural inclined position during sedimentation.
The hindered settling velocity for the spherical particles was verified through sedimentation tests. From these experiments, we found that the hindered settling velocity of the spherical particles is proportional to the porosity with exponent 4.8. This conclusion corresponds to data previously mentioned in the literature.
The main part of this article focused on the hindered settling velocity of significantly non-isometric particles with ratio d/l = 0.3/20. The sedimentation of non-isometric particles up to the ratio d/l = 1/10 can be found in the literature. However, the conclusions mentioned in the literature cannot be used for significantly non-isometric particles due to the different behavior of these particles. In our experiments, we found that the hindered settling velocity of the used significantly non-isometric particles was proportional to the porosity more so than was the case for spherical particles, i.e., the hindered settling velocity of non-isometric particles was proportional to the porosity with exponent 22.1. The great increase in the exponent is the effect of both the shape of the particles, and in particular of the mutual influence that arises from their interweaving and connection.
These results can be applied, for example, to predict the settling velocity of fibers in fresh HPC/UHPC mixtures with negligible yield flow stress. Based on the experimentally determined apparent viscosity of fresh HPC/UHPC mixture, with a given concentration of dosing fibers it is possible to directly determine the settling velocity of the fibers from the proposed correlations, and thus to predict inhomogenous fiber distribution in concrete units due to gravitational sedimentation.