1. 2006 GSK Mathematical Modeling Symposium Modelling Particle Flow Dynamics Using Discrete Element Methods Mark Palmer, Valeriu Damian-Iordache, Pankaj DoshiRob Tuley, John Shrimpton GlaxoSmithKline, UK & USImperial College London Introduction Dry particulate powders are used extensively in many pharmaceutical processes and clinical devices. A computational model of the particles’ physical movement and interaction would be useful for both device design and to further fundamental understanding of particle behaviour. This ‘interaction’ includes both that with other particles and continuum fluids (e.g. particle drag forces). Most existing particle modelling techniques use eulerian methods to track particle concentration through a computational mesh, and statistical approximations to include the effects of particle-particle interactions. However, these techniques are restricted to completely fluidized particle systems (with no enduring particle-particle contacts) and monodisperse (single size) populations. However, most pharmaceutical powder blends are highly polydisperse, with most processes involving at least some non- fluidized densely packed regions. A collaboration between Imperial College London and GSK departments in both the UK and US is attempting to apply lagrangian discrete element methods (DEM) to pharmaceutical powders. A DEM technique tracks individual particles rather than a population concentration, and resolves each inter-particle collision instead of using statistical sampling. Although the method has a greater computational expense than traditional eulerian approaches, it is capable of modelling densely packed polydisperse particle populations with non-fluidized regions such as those found in dry powder inhalers (DPI). Model Framework In a DEM model, particles are initialized and tracked individually rather than using statistical methods. Each particle has a specified size, shape and material properties. Evaluating Forces The acceleration of each particle at any time-point can be evaluated by considering the forces that the particle is subject to. Particle interaction forces (from collisions) are calculated from a number of collision laws (see the ‘Particle Collision’ section). In addition, external forces such as fluid drag and gravity are also applied to each particle. For densely packed particle populations, the presence of these particles will have a significant effect on the fluid flow. Work is on-going to link the numerical solution process of both fluid and particle medium to accurately resolve this behaviour. Particle Trajectory Tracking Particle Collisions A pair of particles are classified as colliding if their volumes overlap (see figure 1). In this situation, various force models can be used to evaluate the forces exerted by each particle on the other based on the area of overlap. The simplest normal force model uses a linear Hookes spring law interaction, but more common are variants of Hertz theory for curved surface contact. Friction is modelled tangential to the contact normal. Other forces such as Van der Waals can be added as required. Figure 3. A possible model development path Figure 1. Illustration of a particle collision The collision time is regarded as finite, with the collision resolved over multiple discrete time-steps. COLLISION PERIOD The trajectory of each particle is resolved from the acceleration over a number of discrete time-points by using common time integration schemes. With full access to the program source code, the project is not restricted to traditional explicit integrators, and a more efficient implicit scheme is being implemented. Recent work at Imperial has shown that the overall accuracy of both explicit or implicit schemes is limited to 2nd order by discontinuities in the applied force produced by particle collisions. t=t 0 t=t 0 +  t t=t 0 +2  t t=t 0 +3  t Application Our work focuses on modelling the particle flow through dry powder inhalers (DPI), specifically the Diskus device. Experimental work has been undertaken to record the dose fluidisation process for simplified optical geometries containing various powders, and it is hoped the current work will be validated against this data. The relatively low number of particles and short timescales (typically 20ms) involved in a single DPI dose evacuation make the simulation feasible for the computationally expensive DEM technique. The capability of the general DEM method extends beyond simplified device prediction: additional particle interaction models can be incorporated to include factors such as humidity effects. Including such factors into the simulation of particulate storage and agglomerate formation or breakup will be able to contribute to the fundamental understanding of these processes. Figure 2. A typical DPI dose evacuation A sequence of experimental images show the evacuation of glass beads from a simplified DPI geometry. The powder is entrained into the airflow (or fluidised) as the air flows from right to left through the U-shaped geometry. It is clear that large areas of densely packed un-fluidized particulate material needs to be simulated to model this application. 1 2 34 5 Computational Expense Although a DEM approach provides a means to model densely packed polydisperse regimes, the method requires a large amount of computer time to complete a simulation. This time scales approximately linearly with the number of particles in the simulation. This means that a single DPI dose evacuation simulation requires a parallel software code, running on parallel computer hardware. A typical pharmaceutical powder blend contains a significant proportion of particle ‘fines’, or very small particles. Although the mass fraction of fines in a blend may be small, their size means that the actual number of individual particles remains large. The problem is addressed by introducing a resolvable DEM particle cut-off size, and including the effects of particles below this size (such as their contribution to bulk cohesion) by modification of the larger particle interaction laws. While the DEM technique is limited at present to small particle populations, it returns a highly detailed picture of the physical particle interaction in a process. The model also provides an intuitive framework for continuous extension and improvement, and simulations can be scaled in line with the computer resources available. A possible model development path including and beyond the current work is illustrated in figure 3. Increasing Accuracy eulerian statistical methods eulerian method replaced by a DEM algorithm with spherical particles and simple linear particle interaction addition of more complex hertzian particle interaction laws surrounding fluid medium included and solved using turbulent CFD. Fluid-particle effects such as drag are evaluated. additional model developed to include the effect of particle below the size cut-off more complex particle interaction models used to include factors such as humidity introduction of non-spherical particle shapes. fluid flow fully resolved around particle surfaces using particle surface patches Increasing Computational Expense per Particle SIMPLIFIED DEVICE PREDICTION AGGLOMERATE BREAKUP MODELING