(712c) Numerical Modeling of Two-Component Granulation

Granulation of pharmaceutical powders generally involves the mixing of an active pharmaceutical ingredient (API) and an excipient in the presence of a liquid binder. To apply population balance models, one must adapt the usual one-component algorithms to a bivariate distribution that takes into consideration both the size and composition of the granules. The numerical challenges of this problem are quite significant and the extension of one-component methods are not straightforward. These difficulties are magnified when one considers a realistic granulation environment in which the population balance equation (PBE) must be solved alongside with transport equations for momentum and heat in three-dimensional space. Though our work currently focuses on the two-component PBE alone, our goal is to perform realistic large-scale simulations of a fluidized bed granulator. Our short-term goal, and the focus of this work, is in developing numerical solutions of two-component PBE that are both sufficiently accurate and sufficiently fast to apply to real systems. In this study, we apply three different techniques with complementary advantages and limitations:

(i)         Direct solution of the discrete, two-component PBE. This approach is rigorous and detailed but is computationally demanding and can only be used to track the very early stages of granulation

(ii)        Constant-Number Monte Carlo (cNMC). This method allows the computation of the bivariate distribution over arbitrarily long times. Its chief disadvantage is that is not well-suited for systems that involve space and time gradients. This is due to the computational cost associated with increased dimensionality in space, and also due to a fundamental mismatch between event-driven Monte Carlo and time-driven process simulators.

(iii)       Direct Quadrature Method of Moments (DQMOM): this methodology has proven to be very efficient in one-component systems and is currently the only viable option for interfacing the PBE with fluid dynamics. The method is applicable to two-component systems but such extensions are not straightforward, nor can they be implemented uniquely. Therefore, experimentation with DQMOM requires validation against known solutions.

(iv)       Theory. Recent work from our group has resulted in the formulation of a theory of bicomponent aggregation, which, for special classes of kernels is capable of predicting the analytic form of the compositional distribution at long times. While exact predictions are not possible for all granulation kernels, theory is valuable in establishing known solutions and a basis for the interpretation of results.

To keep the formulation general, the aggregation kernel is expressed in the form K₁₂=k₁₂Y₁₂, where k₁₂ depends on the size of the aggregating particles (granules) and Y₁₂ depends on their composition. For k₁₂  we use the kinetic theory of a granular flow: k₁₂=bY₁₂d^g₄₃(d₁+d₂)²(1/d₁³ + 1/d₂³)^0.5

where d_i is the diameter of particle i, which is assumed to be spherical. For Y₁₂ we have chosen the expression Y₁₂=exp[-a_AB(c₁+c₂-2c₁c₂)]

where a_AB is a parameter and c_i is the mass. By choosing the value of a_AB we can simulate a wide range of behaviors. With a_AB<0, collisions between granules with dissimilar composition (solute-rich with excipient-rich) are favored, giving rise to enhanced mixing between components. With a_AB>0, the opposite is true: collisions between similar components are favored leading to poor mixing and possibly segregation of components. Finally, with a_AB=0 we recover the composition-independent kernel for which theory provides solutions.

We present results of simulations for various initial conditions in terms of amount of API and size ratio of the API and excipient particles. We find that Monte Carlo simulations with N=1000 to 10000 provides very good agreement with the rigorous (discrete) PBE and also with the predictions of the theory. Specifically, we find that the distribution of API in granules of mass v_i is a Gaussian function whose mean is equal to the overall mass fraction of API in the granulator and whose standard deviation is inversely proportional to the square root of v_i. Thus we conclude that the cNMC method produces accurate solutions over very long times.  We present further results with both positive and negative values of the parameter a_AB and discuss the conditions under which the DQMOM method provides accurate solutions to the bicomponent problem.