Mathematical modelling of granulation processes : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Physics at Massey University, Palmerston North, New Zealand
Granulation is an industrial process where fine particles are bound together into larger granules. The process has numerous applications including the manufacture of pharmaceuticals and the production of cosmetics, chemicals, detergents and fertilisers. This thesis studies aspects of wet granulation which involves the application of a viscous binder, usually in the form of a spray, to an agitated bed of powder particles. Individual powder particles may adhere together, joined by small quantities of binder fluid called liquid bridges. By a process of collision and adherence additional particles may join the newly formed agglomerates. Agglomerates may also coalesce together which is a process that leads to granule formation. On the completion of this process, granules are typically dried.This thesis studies wet granulation on three different levels. First, micro-level investigations of liquid bridges between two and three particles are performed. For the two-particle case, the fluid profile of static (stationary) and dynamic (moving) liquid bridges is investigated. For the static case, a numerical solution to the Young-Laplace equation is obtained; this relates the volume of binder fluid to liquid bridge properties such as the inter-particle force. An analytic solution is also obtained, providing the liquid bridge profile in terms of known mathematical functions. For both solutions, the radii of the (spherical) primary particles may be different. The dynamic case is then studied using the Navier-Stokes equations with the low Reynolds number approximation. The motion of the approaching particles is shown to be damped by the viscosity of the liquid bridge. Static liquid bridges between three equally sized primary particles are then studied. Symmetry of the problem is used to obtain a numerical solution to the Young-Laplace equation. Liquid bridge properties are calculated in terms of the binder fluid volume. Experimental agreement is provided.Secondly, a model to estimate the stickiness (fractional wet surface area) of agglomerates is proposed. Primary particles are approximated as spheres and are added one at a time in a closely packed arrangement. The model includes parameters to control the inter-particle separation distance and the fluid saturation state. Computational geometry is used to obtain results which relate the number of particles and the volume of binder fluid to the stickiness of the agglomerates.Finally, a population balance model for wet granulation is developed by extending an earlier model to incorporate the effects of binder fluid. Functions for the inter-particle collision rate and drying rate are proposed, including functions which are derived from the geometric model, described above, for the case of maximum particle consolidation. The model is solved numerically for a range of coalescence kernels and results are presented which show the effect of binder volume and the drying rate.