At the beginning of the second half of the twentieth century, Proudman and Pearson (J. Fluid. Mech.,2(3), 1956, pp.237–262) suggested that the functional form of the drag coefficient (CD) of a single sphere subjected to uniform fluid flow consists of a series of logarithmic and power terms of the Reynolds number (Re). In this paper, we will explore the validity of the above statement for Reynolds numbers up to 106 by using a symbolic regression machine learning method. The algorithm is trained by available experimental data and data from well-known correlations from the literature for Re ranging from 0.1 to 2×105. Our results show that the functional form of CD contains powers of log(Re), plus the Stokes term. The logarithmic CD expressions can generalize (extrapolate) better beyond the training data than pure power series of Re and are the first in the literature to predict with acceptable accuracythe onset of the rapid decrease (drag crisis) of CD at high Re, but also to follow the right behaviour towards zero Re. We also find a connection between the root of the Re-dependent terms in the CD expression and the first point of laminar separation. The generalization behaviour of power-based drag coefficient equations is worse than logarithmic-based ones, especially towards the zero Re regime in which they give non-physical results. The logarithmic based CD correctly describes the physics from the low Re regime to the onset of the drag crisis. Also, by applying a minor modification in the logarithmic based equations, we can predict the drag coefficient of an oblate spheroid in the high Re regime.@en

In full-scale drinking water production plants in the Netherlands, central softening is widely used for reasons related to public health, client comfort, and economic and environmental benefits. Almost 500 million cubic meters of water is softened annually through seeded crystallisation in fluidised bed reactors. The societal call for a circular economy has put pressure on this treatment process to become more sustainable. By optimising relevant process conditions, the consumption of chemicals can be reduced, and raw materials reused. Optimal process conditions are feasible if the specific crystallisation surface area in the fluidised bed is large enough to support the performance of the seeded crystallisation process. To determine the specific surface area, crucial variables including voidage and particle size must be known. Numerous models can be found in the literature to estimate the voidage in liquid-solid fluidisation processes. Many of these models are based on semi-empirical porous-media-based drag relations like Ergun or semi-empirical terminal-settling based models such as Richardson-Zaki and fitted for monodisperse, almost perfectly round particles. In this study, we present new voidage prediction models based on accurate data obtained from elaborate pilot plant experiments and non-linear symbolic regression methods. The models were compared with the most popular voidage prediction models using different statistical methods. An explicit model for voidage estimation based on the dimensionless Reynolds and Froude numbers is presented here that can be used for a wide range of particle sizes, fluid velocities and temperatures and that can therefore be directly used in water treatment processes such as drinking water pellet softening. The advantage of this model is that there is no need for applying numerical solutions; therefore, it can be explicitly implemented. The prediction errors for classical models from the literature lie between 2.7 % and 11.4 %. With our new model, the voidage prediction error is reduced to 1.9 %.@en

The twenty first century is the century of data. Machine learning data and driven methods start to lead the way in many fields. In this contribution, we will show how symbolic regression machine learning methods, based on genetic programming, can be used to solve fluid flow problems. In particular, we will focus on the fluid drag experienced by ellipsoidal and spherocylinder particles of arbitrary aspect ratio. The machine learning algorithm is trained semisupervised by using a very limited amount of data for a specific single aspect ratio of 2.5 for ellipsoidal and 4 for spherocylindrical particles. The effect of the aspect ratio is informed to the algorithm through what we call previous knowledge, for example, known analytical solutions in certain limits, or through interbreeding of different flow solutions from the literature. Our results show good agreement with literature results, while they are obtained computationally faster and with less computing resources. Also, the machine learning algorithm discovered that for the case of prolate spheroids, the difference between the drag coefficients perpendicular and parallel to the flow in the high Reynolds number regime only depend on the aspect ratio of the geometry, even when the individual drag coefficients still decrease with increasing Re.@en

In this paper, we present a number of key numerical methods that can be used to study elongated particles in fluid flows, with a specific emphasis on fluidised beds. Fluidised beds are frequently used for the production of biofuels, bioenergy, and other products from biomass particles, which often have an approximate elongated shape. This raises numerous issues in a numerical approach such as particle-particle contact detection and the accurate description of the various hydrodynamic forces, such as drag, lift, and torque, that elongated particles experience when moving in a fluid flow. The modelling is further complicated by a separation of length scales where industrial flow structures that can extend for many metres evolve subject to solid-solid and solid-fluid interactions at the millimetre scale. As a result, it is impossible to simulate both length scales using the same numerical approach, and a multiscale approach is necessary. First, we outline the direct numerical simulation (DNS) approach that may be employed to estimate hydrodynamic force closures for elongated particles in a fluid flow. We then describe the key aspects of a CFD-DEM approach, which can be used to simulate laboratory scale fluidisation processes, that must be addressed to study elongated particles. Finally, we briefly consider how current industrial-scale models, which concretely assume particle sphericity, could be adapted for the simulation of large collections of elongated particles subject to fluidisation.@en

The rheology of suspensions of high-inertia (or granular) non-spherical particles characterized by high particle Stokes and Reynolds numbers is rarely investigated. In this study, we investigate the rheology of suspensions of inertial rod-like particles of aspect ratio 4 subjected to shear flow. In particular, the effect of fluid medium (air, water) against dry granular simulations on the developed stresses is assessed. CFD-DEM simulations are performed for a periodic shear box for a range of shear rates and volume fractions of particles. The dependence of rheological properties like shear stress, normal stress difference, pressure and relative viscosity on volume fraction, shear rate, granular temperature and the particle orientation are discussed. These results provide insight into the macroscopic rheology of suspensions of rods and demonstrate that the effect of particle shape and surrounding fluid cannot be completely ignored. Air as a fluid medium shows similar scaling as compared to dry granular simulations, but the stress values are generally lower. We observe drastic change in both scaling and values for water as fluid medium. In all cases, the rods show strong alignment in the direction of shear. This study can be further extended to develop stress closures for use in Eulerian flow models.@en

For an accurate prediction of the porosity of a liquid-solid homogenous fluidized bed, various empirical prediction models have been developed. Symbolic regression machine learning techniques are suitable for analyzing experimental fluidization data to produce empirical expressions for porosity as a function not only of fluid velocity and viscosity but also of particle size and shape. On the basis of this porosity, it becomes possible to calculate the specific surface area for reactions for seeded crystallization in a fluidized bed.@en

Self-propelled nanofluids (SPNFs), are suspensions that contain active particles, that self-propel by converting some form of energy to mechanical work. This theoretical investigation considers the heat transfer mechanisms that may exist in an SPNF. Equations describing the effective mass diffusivity of spherical and rod-shaped particles were taken from the literature and used in analysis of particles of different shapes, aspect ratios, swimming velocities and suspended in water and in ethylene glycol. The analysis showed that the effective mass diffusivity of the particles was up to three orders of magnitude higher than the thermal diffusivity of the solvent. The enhancement of the mass diffusivity leads to a significant enhancement of thermal conductivity of up to an order of magnitude higher compared to that of the pure solvent. It is further discussed that in SPNFs, dispersion and clustering of particles and active turbulence mechanisms may add extra enhancement to the thermal transport. Recent experimental investigations supporting our findings are discussed. Combining this enhancement in thermal transport with the reported reduction of the viscosity observed for an SPNF consisting of Artificial Bacterial Flagella (ABF) particles, will help to create a highly efficient coolant. This can help to reduce energy consumption in a wide variety of the economic sectors, especially in the computing and data storage.@en

Nanostructured phase change materials (NEPCM) colloidal suspensions, got the attention from the scientific community due to their promising thermal properties that allow for faster solidification, and melting times. However, most of the experimental investigation shows the opposite, that the melting and freezing times are increased as the volume of the particles is increased. This investigation will be the first in the literature to include the mass transport of the particles, for the case of melting of NEPCM that will help understanding better the melting process of the NEPCM. In this paper, the development of the solid-liquid interface, the distribution of the nanoparticle profiles, as well as the development of the thermal convection, will be investigated for the case of melting of nanostructured phase change materials (NEPCM) colloidal suspensions, inside a rectangular cavity. The numerical model is based on the one-fluid-mixture approach combined with the single-domain enthalpy-porosity model for phase change. The linear dependence of the liquids, and solidus temperatures with the concentration of the nanoparticles was assumed. The NEPCM consists of water and copper nanoparticles, the nanoparticle size was selected to be 5 nm and 2 nm. The suspension was melted inside a rectangular cavity, heated from the left side. It was observed that for the case of dp = 2 nm, as the mass fraction of the particles increases the solid-liquid interface changes from a planar to unstable morphology during melting. Furthermore, as the mass fraction of the particles increases the temperature of the suspension decreased similarly to experimental observations. Also, I found that the rate of the rejection of the particles and the particles size plays an essential role in the development of the liquid fraction, concentration field, and the resulted thermal convection.@en