Monitoring, modelling, and control of particle size distributions of final active pharmaceutical ingredients in air jet milling processes

Particle size reduction is an important step in the design, development, and processing of active pharmaceutical ingredients (API). Amongst the variety of breakage equipment available, air jet mills are much favoured in the pharmaceutical industry due to their non-degrading and non-contaminating nature. Jet mills have a simple design and operation. They consist of a cylindrical (or an elliptical) chamber in which gas is blown through nozzles angled tangentially to the wall. The gas creates a vortex into which the particles are fed by a vacuum injector. The inter-particle collisions and particle-wall collisions lead to particle breakage. As the particles get smaller, they experience a higher radial force which drags them towards the mill outlet. The jet mill operation is controlled by manipulating three variables: (i) the grinding pressure which dictates the fluid energy transferred to the particles in the mill, (ii) the solid feed rate, which dictates the mill throughput and the amount of particles present in the mill, and (iii) the injector pressure, which create a vacuum to suck the particles fed through hoppers to the milling chamber. Out of the three variables, the injector pressure does not have much influence on the milling efficiency and is always maintained at a pressure slightly ($\sim 0.1$ bar) above the grinding pressure. Thus, the operation of jet mill is dictated by only two variables. Despite the ease of operation, describing the jet mill using available mathematical modelling approaches is extremely challenging. In this thesis, two commonly used approaches for modelling particulate processes are considered: population balance models and discrete element models.

The first objective of this thesis is to determine the best method to solve a continuous population balance model involving only breakage. Three commonly used sectional methods are considered: the fixed pivot, the moving pivot, and the cell average method. On coarse grids, the moving pivot technique outperforms the other two methods. However, the moving pivot method is computationally very expensive. The performance of all the methods increases with higher discretization, but quickly converges to a constant value. At higher discretization, the cell average method performs better than the two other methods.

The second objective of the thesis is to determine whether unique parameter estimation is possible for the PBM of a jet mill. The identifiability of the model is assessed using the profile likelihood approach. It is shown that the breakage and classification kernels are impossible to identify when the mill is operated in the semi-continuous mode with particle size being measured only at the end of the milling batch. The identifiability of the model improves when the particle size is recorded continuously. However, the confidence intervals of the identified parameters remain large. To further assess the effect that the parametric uncertainties can have on the predictive capabilities of the model, an analysis of three uncertainty propagation techniques is carried out. The sigma-point approach provides the best compromise between efficiency and accuracy for the uncertainty propagation when compared to linearization and polynomial chaos expansion methods. In all cases, it is seen that the model prediction from the continuous population balance of the cone mill has a 95\% confidence interval of $\pm$100 $\mu$m, which is too large in relation to the predicted output of around 300 $\mu$m. This further shows the importance of accurate and unique parameter estimation.

The third objective is to describe the fluid and solid dynamics within the jet mill using computational fluid dynamics coupled with the discrete element method. A two-way coupling is used to consider the effect of particle phase on the fluid. Three different loading conditions are assessed along with two operating pressures. It is shown that the particle phase decelerates the fluid in the jet mill. The higher the particle loading, the lower the fluid velocity in the mill. The fluid field in turn affects particle motion. Thus, at higher particle loading the average particle velocity and consequently, the average collision velocity is reduced. However, the number of collisions increases with increasing particle loading. Similar observations are made from the two operating pressures, higher pressure leads to increase in the fluid, particle, and also collision velocity. Moreover, an increase in pressure also increases the total number of collisions. In all cases, the particle-particle collisions are the primary driver for breakage.