Derivation of drag and lift force and torque coefficients for non-spherical particles in flows

https://doi.org/10.1016/j.ijmultiphaseflow.2011.09.004Get rights and content

Abstract

This paper derives and validates a new framework to predict the drag and lift coefficients as well as the torque coefficients for four non-spherical particle shapes in a flow with a wide range of flow Re and rotational Re numbers. Correlations are proposed for the drag force, the lift force, the pitching torque, and the torque caused by the rotation of the particle. Each of the correlations depends on Re number, the dimensionless rotation and the angle of incidence between the particle and the direction of the local fluid velocity. The fit parameters in the correlations for each of the particle shapes are determined by performing a large number of “true” DNS simulations of four different types of particles. The true DNS simulations are carried out with an improved mirroring immersed boundary method. The resulting correlations for the forces and the torques are suitable to be used in Eulerian–Lagrangian simulations, where an accurate prediction of the forces and torques is required to determine the motion of the particles.

Highlights

► We use DNS to model the flow around four types of non-spherical particles. ► The results are compared to literature correlations. ► New models are proposed for drag, lift and the two torques for the particles. ► The models fit very well with the simulation results. ► These models can be used in large-scale gas–solid computations.

Introduction

The ability to predict the behaviour of turbulent gas–solid flows is vital for the successful design and determination of optimum operating conditions of numerous industrial applications, e.g. cyclone separators, fluidised beds, dust collectors, and pulverised-coal combustors to name a few. The dynamics of these type of systems can be investigated through experiments or through numerical simulations. The low cost and large amount of data and insight that can be obtained make the numerical approach very convenient option. Still, performing large scale numerical study of complex multiphase flow requires some assumptions and empirical data describing the interactions between the fluid and the particles.

So far, nearly all studies performed on the gas–particle flows model particles as perfect spheres. This assumption is very convenient due to its simplicity, the fact that the behaviour of spheres is well known, and the availability of a number of models to describe the interaction with fluid flow. Dependence of the drag coefficient of a sphere on the Reynolds number can be found for example in Schiller and Naumann (1933). Also behaviour of rotating spheres and spheres moving in a shear flow have been studied with additional correlations for Magnus and Saffman forces available in Niazmand and Renksizbulut, 2003, Kurose, 1999, Dennis et al., 1980.

Nevertheless, a vast number of the applications deals with non-spherical particles, which makes analysis of these type of flows more complicated. Spheres can be described by a single characteristic value, i.e. the diameter, whereas non-spherical particles require more parameters. Even very regular shapes, like ellipsoids or fibres, are described by at least two parameters. Moreover, the particles can have varying orientation with respect to the flow, what additionally complicates the description of their behaviour. Besides the drag force, a non-spherical particle also experiences a transverse lift force along with pitching and rotational torques.

Even though most of the papers on gas–solid flows focus on spherical particles, the effects of non-sphericity had been addressed by some researchers. In order to account for the deviation from the idealised spherical shape, a so-called “sphericity factor”, Φ, has been introduced (Wadell, 1934, Kunii and Levenspiel, 1991). Sphericity is defined as the ratio of the surface area of a sphere with equal volume as the non-spherical particle over the surface area of the non-spherical particle. By definition, the sphericity factor is less than or equal to one, where one corresponds with a sphere. In most engineering handbooks (e.g. Crowe, 2006) the drag of a non-spherical particle is estimated by using correlations for spherical particles and modified to take into account the sphericity factor.

Using sphericity to describe non-spherical particles may give promising results, nonetheless it is far from ideal. For instance, the same value of sphericity can be obtained for a needle like prolate ellipsoid and for a disc, while their behaviour in the flow will be different. Moreover, the sphericity does not account for the orientation of the non-spherical particle. In order to introduce orientation dependency in drag correlations, some researchers, like Hölzer and Sommerfeld (2008), use two additional factors: the sphericity determined in the lengthwise direction and one in the crosswise direction, making the effective sphericity orientation dependent.

Other ways to describe the shape of the particle are proposed by Rosendahl (2000) or by Loth (2008a). Rosendahl uses the super-elliptic function to describe the particle shape and to predict the drag at two extreme orientations, i.e. aligned with the flow, and at 90° relative to incoming fluid velocity. On the other hand Loth describes the particle by its aspect ratio, which is applied to calculate shape correction factors both in parallel and cross-wise directions. The effects of orientation can be also included by modifying the reference area in force coefficient expressions. The most complete overview of the existing methods for analysing non-spherical particles can be found in a paper by Mando and Rosendahl (2010). So far, the majority of the research has been focused on determining the drag coefficient, while the secondary motion resulting from the lift and torques has received very little attention.

The current paper shows the results of direct numerical simulations of the flows past four different non-spherical particles presented in Table 1. The obtained forces are then used to design shape-specific correlations, that describe the interactions between the fluid and the particles. The equations can be used as a base of large scale analysis of complex flows with non-spherical particles.

Section snippets

Forces on particles

Few closed models describing the motion of a non-spherical particle in the fluid are available. In a Lagrangian framework the translation motion of particles can be described by Newtonian equations of motion (Yin et al., 2003):mpDvpDt=FD+Vp(ρp-ρf)g+FPG+FVM+FLwhere mp is the particle mass, Vp is the particle volume, ρp is the particle density, ρf is the fluid density, and vp is the translational velocity of the particle centre of mass. The forces acting on the particle are given on the

Numerical framework

Because of the large number of simulations required, a computationally effective and efficient framework is desired to deal with flows including non-spherical particles. Moreover, the framework should be able to handle rotating particles. Throughout the years, various methods coupling the particles with surrounding fluid have been developed. Among the oldest ones is the arbitrary-Lagrangian–Eulerian method (Hu, 1996), where a two-dimensional unstructured grid is created around the bodies and

Simulation set-up

A true direct numerical simulation (DNS) framework has been used to determine the drag, lift and torque of the bodies shown in Table 1. A triangulated representation of the bodies is introduced in a domain with size 20dp × 20dp × 10dp for low Reynolds number Re < 1 and a cubical domain 10dp × 10dp × 10dp for high Re number simulations. The set-up is illustrated in Fig. 5.

A uniform flow with the velocity U = 1.0 m/s is set along the positive x axis. The fluid density is ρ = 1 kg/m3. A full slip (i.e. no

Results and discussion

True DNS of the flow past each particle type is performed in order to obtain shape-specific drag, lift and torque characteristics as a function of Reynolds number and angle of incidence. The flow is calculated with enforcing a no-slip boundary condition at the particle surface by the mirroring immersed boundary condition presented in Section 3. The length of simulations allows to obtain steady state solution for low Re or constant averages in case of unstable flows. A sample result is

Conclusions

The behaviour of the interaction of non-spherical particles with a fluid flow is a complex phenomenon, even for axisymmetric particles in a uniform flow. Shape has a great influence on the behaviour of the particle, not only by changing the values of the experienced forces and torques but also by shifting the Reynolds number at which the transition to unsteady flow occurs. The flow field also strongly depends on the angle of incidence between the particle and the incoming fluid velocity.

Even

Acknowledgements

The authors are grateful to the Engineering and Physical Sciences Research Council (EPSRC) for their financial support (Grant No. EP/G049262/1) and the Jan Dzienisiewicz trust for providing M. Zastawny with a bursary.

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