# Computational Modelling

### Introduction

A wide range of industrial systems depict behaviour at the length scale of interest that is dominated by interactions of discrete material regions. In the minerals industry, for example, there are many processes which may be modelled as discrete particle distributions within a fluid medium. Fluid-particle mixtures constitute a rheologically complex fluid which is transported through a dynamic porous bulk. Mineral processing machines, such as tumbling mills and flotation cells, are presently described mainly by phenomenological models. The preponderance of such models is strongly coupled to the lack of in-situ measurements which are needed to elucidate the physical parameters that constitute the underlying mechanism. Clear understanding of the mechanistic nature is crucial to the efficiency and overall optimisation of mineral plants.

The major advantage of the computational frameworks like the Discrete element method (DEM), Computational fluid dynamics (CFD) and Smooth particle hydrodynamics (SPH) is the ability to approximate the mechanical environment with the potential to discern meaningful in-situ measurements and behaviour. Numerical simulation is employed in a complimentary manner with other in-situ measurement tools like Positron emission particle tracking (PEPT) and X-ray imaging.

### Whats is DEM?

The Discrete Element Method (DEM) was developed by Cundall and Strack[1] for analysing quasi-static soil mechanics problems. The DEM was revolutionary in that it modelled the individual particles in a discrete system separately and thus represented the first truly discontinuous numerical model. The particles were originally modelled as rigid, circular discs that interacted at contacts only, but DEM has been extended to three dimensional and non-spherical objects by several researchers [2]. Rigid spherical particles as implemented in the commercial software package EDEM developed by DEM Solutions and Particle Flow Code (PFC) developed by ITASCA are used to perform the simulations of an experimental tumbling mill. Contact between particles is modelled using the soft-contact approach wherein contacting particles are allowed to overlap one another at the point of contact. A contact force law is used to relate the relative overlap between the particles to a restoring contact force. The most commonly used contact force law is the viscous damping model. A stiff numerical spring is applied in the normal direction at the contact point. The undamped restoring spring force in the normal direction is calculated as the product of the relative particle overlap and the spring stiffness according to Hooke's law. A second numerical spring, the shear spring, is applied in a direction orthogonal to the normal spring to simulate the frictional forces at the contact. The maximum frictional force cannot exceed the limiting frictional force governed by the Mohr-Coloumb Law. Energy dissipation to non-frictional mechanisms is modelled using numerical dashpots that oppose the contact forces. Newton's second law is used to model the rigid body motion of the particles arising from the contact and gravitational forces acting on the particle. The assumption of rigid particles is valid when the majority of the deformation in the system occurs along interfaces. Thus, the DEM is highly suited for the analysis of granular flow, where the deformation of individual particles has little effect on the mechanical behaviour of the system. The dynamic behaviour of a particle system is analysed by discretising the time domain. The assumption of infinitesimal displacement theory is valid if the time-step is appropriately small. The particle displacements during each time-step are approximated using a conditionally stable, explicit, central difference time-stepping algorithm. The use of an explicit time-stepping algorithm makes the simulation of a system composed of thousands of particles computationally feasible.

### Collaborators

Much of the DEM and CFD research has been focussed on mineral systems like tumbling mills. The Centre for minerals research (CMR) housed in the department of chemical engineering at UCT is a key collaborator in this work.

### What is CFD?

Computational Fluid Dynamics (CFD) is the process of modelling fluid flows by the numerical solution of the Navier-Stokes equations for viscous flows or the Euler equations for inviscid flows. Other equations are also solved simultaneously in cases like multiphase flows in tumbling mills that depict non-Newtonian behaviour.

The standard numerical approach is to discretise the spatial domain into small cells to form a volume mesh or grid. An appropriate algorithm then solves the governing partial differential equations of motion across the cells. The choice of cell geometry is very problem specific ranging from regular triangular cells to non-uniformly varying tetrahedrons.

### What is SPH?

Smooth Particle Hydrodynamics (SPH) is a mesh-free method employed typically in function interpolation and the solution of partial differential equations. Despite the name, SPH can be used to solve non-hydrodynamic problems too.

Unlike conventional grid-based techniques, in SPH the domain is discretised by introducing SPH particles, which are free to roam about the function domain. These particles need not have any physical interpretation, serving solely as carriers of field variable information and their derivatives. Associated with any SPH technique is a kernel function, in terms of which all functions and their derivatives at the location of an SPH particle can be expressed as weighted sums over those neighbouring SPH particles within a finite support radius.

The power of SPH is realised in problems of high-deformation, where grid-based techniques fail. However, SPH is not without its limitations. Boundary effects and the imposition of boundary conditions are the subject of ongoing research, leading to numerous techniques and correction schemes.

### Validation of numerical models

The computational demands and lack of sound experimental verification have limited the value of DEM and CFD techniques in many industries. This work seeks to fill the vital gap linking computational results to rigorous experimental data. It is only with validation that any confidence can be given to the predictive capability of such computational tools, especially when the predictive range lies outside the range over which the existing semi-empirical models were developed and tested. In-situ measurement tools like Positron emission particle tracking (PEPT) and X-ray imaging allow, for the first time, a detailed investigation into aspects such as contact models, energy distributions and flow.

### Radiation Transport

In-situ measurement techniques like PEPT are based on the principle of positron annihilation. Simultaneous detection of the two gamma rays in an array of detectors (a PET “camera”) defines a line along which the annihilation between positron and an electron occurred. Detection of a few such events in a very short time interval allows the position of the tracer particle to be triangulated in three dimensions. Location in space of the tracer particle depends on the speed and activity of the tracer particle, and the attenuating environment that defines the system under study. Complex environments with high speed tracers can lead to unpredictable transport of the resulting gamma rays. To better understand radiation transport in such scenarios computational modelling via monte carlo simulation of the decay and transport process is studied.

### References

- P.A. Cundall and O.D.L. Strack (1979). A discrete numerical model for granular assemblies. Geotechnique, Vol. 29, No. 1, pp 47-65.
- P.W. Cleary, R. Morrison and S. Morrell (2003). Comparison of DEM and experiment for a scale model SAG mill. International Journal of Minerals Processing, Vol. 68, pp. 129-165.