One-dimensional Population Balance Model Research Articles

Model development and calibration of two-dimensional population balance model for twin-screw wet granulation based on particle size distribution and porosity

Population balance models (PBMs) have been widely studied for the simulation of continuous twin-screw wet granulation. Due to the limited data availability and computational efforts, one-dimensional PBM is the most popular application to the pharmaceutical industry. While multi-dimensional PBMs can predict multiple granule quality attributes, model development and calibration has been the big challenges. This paper presents a two-dimensional compartmental PBM which compute granule size and porosity. Granulation experiments were performed for two formulations and two process settings, followed by granule characterizations. In PBM development, consolidation was newly introduced, whereas the aggregation and breakage kernels were based on the previous model. Calibration results proved that the model can simulate granule size and porosity simultaneously for different formulations and process settings. Some cases show porosity increase that needs to be considered in the future. Further experiments and model development can increase the applicability of the two-dimensional PBM for the pharmaceutical industry.

Simplifying population balance models to promote broader use in industry

The approach outlined in this paper takes advantage of the existence of stationary states and similarity solutions to reduce a one-dimensional population balance model with particle volume as the internal dimension from a partial integro-differential equation to a single ordinary differential equation for the evolution of the mean size. The mechanisms of accretional growth, collisional growth and breakage, either individually or in all possible combinations, are considered. Power-law rate kernels are employed and a self-similar fragment distribution is assumed for the breakage term. When expressed in terms of the evolution of the scaled number-mean particle volume, the model parameters consist of kernel orders and the characteristic times associated with each mechanism. Equations for the scaled moments of the stationary states and similarity solutions are exhibited for all possible mechanism combinations as are equations for the trajectories of scaled number-mean particle volume. Stability of the stationary states and similarity solutions is addressed in an appendix. It should be clear that when this level of simplification is possible, the subject of population balance modeling becomes considerably easier to teach while its use becomes much easier to promote in industry. The use of this simplified approach for extraction of model parameters from experiment is demonstrated in an example, followed by a brief discussion of using the results for scale up. To close out the paper, the same approach, but now in two internal dimensions (particle volume and surface area), is used to demonstrate the capability of the model to unify a set of operating conditions for carbon black manufacture across multiple grades and two different types of synthesis reactors. This is done by extracting the characteristic coalescence time at multiple temperatures and correlating the results in an Arrhenius plot. The model matches known values of primary particle size, and predicts a key indicator of the product’s fractal aggregate structure, namely the number of primary particles per aggregate. The model results are then pushed to yield insight into the nature of the possible collision and coalescence mechanisms.

Simulating the wet granulation of TiO2 photocatalyst in fluidized bed: Population balance modelling and prediction of coalescence rate

The major focus of this study, the second part of a series, is to simulate the net granulation process of TiO2 powder using 1-D discretized population balance modelling (PBM) and Size-Independent Kernel (SIK) coalescence model. In this, sequential and non-sequential modelling procedures have been used.Tested one-dimensional population balance model and coalescence-only, size-independent kernel provide a good prediction of real TiO2 particle size distributions (PSDs). The coalescence rate is quantified for both modelling procedures and formulation system used. Reduced-order population balance equation (PBE) and quite simple, fully empirical coalescence-only kernel successfully describe and predict real TiO2 PSDs. This paper reveals the meso-scale study of the wet granulation process for the not previously granulated powder system.

Integrating crystallization with experimental model parameter determination and modeling into conceptual process design for the purification of complex feed mixtures

In this study, crystallization process simulation combined with experimental model parameter determination at lab-scale is investigated in order to allow the integration of crystallization unit operation into conceptual process design for the purification of complex mixtures and possibly assist in formulation.A one-dimensional population balance model is combined with experiments, which are selected and carried out as a typical example for an industrial fermentation broth (e.g. vanillin), focusing on determination of solubility and growth kinetics as well as kinetics of agglomeration and breakage. Model parameter determination and model validation show that the named effects are adequately described by the model. Hence, model-based process design of purification by crystallization and particle formation enabling integration into formulation considering relevant effects regarding a complex feed mixture becomes possible within a conceptual process design. Further applications are under consideration.

Experimental study and modeling of particle drying in a continuously-operated horizontal fluidized bed

Multistage fluidized beds are frequently used for product drying in industry. One advantage of these fluidized beds is that they can achieve a high throughput, when operated continuously. In this study, γ-Al2O3 particles were dried in a pilot-scale horizontal fluidized bed, without considering any comminution effects. For each experiment, the particle moisture content distribution and residence time distribution were determined. To take into account particle back mixing in our experiments, a one-dimensional population balance model that considers particle residence time was introduced into a fluidized bed-drying model. Experimental particle residence time distributions were reproduced using a tank-in-series model. Subsequently, the moisture content distribution was implemented, as a second dimension to the population balance in this model. These two-dimensional simulations were able to describe the experimental data, especially the spread in the residual particle moisture distribution, much more accurately than one-dimensional simulations. Using this novel two-dimensional model, the effects of different operating parameters (process gas temperature, solid feed rate, superficial air velocity) on the particle moisture content distribution were systematically studied.

Model-based analysis of a twin-screw wet granulation system for continuous solid dosage manufacturing

Implementation of twin-screw granulation in a continuous from-powder-to-tablet manufacturing line requires process knowledge development. This is often pursued by application of mechanistic models incorporating the underlying mechanisms. In this study, granulation mechanisms considered to be dominant in the kneading element regions of the granulator i.e., aggregation and breakage, were included in a one-dimensional population balance model. The model was calibrated using the experimentally determined inflow granule size distribution, and the mean residence time was used as additional input to predict the outflow granule size distribution. After wetting, the first kneading block caused an increase in the aggregation rate which was reduced afterwards. The opposite was observed in case of the breakage rate. The successive kneading blocks lead to a granulation regime separation inside the granulator under certain process conditions. Such a physical separation between the granulation regimes is promising for future design and advanced control of the continuous granulation process.

The space time CE/SE method for solving one-dimensional batch crystallization model with fines dissolution

This article is concerned with the numerical investigation of one-dimensional population balance models for batch crystallization process with fines dissolution. In batch crystallization, dissolution of smaller unwanted nuclei below some critical size is of vital importance as it improves the quality of product. The crystal growth rates for both size-independent and size-dependent cases are considered. A delay in recycle pipe is also included in the model. The space–time conservation element and solution element method, originally derived for non-reacting flows, is used to solve the model. This scheme has already been applied to a range of PDEs, mainly in the area of fluid mechanics. The numerical results are compared with those obtained from the Koren scheme, showing that the proposed scheme is more efficient.

Modeling aggregation kinetics of fluidized bed spray agglomeration for porous particles

The present work develops a comprehensive one-dimensional pure agglomeration population balance model for simulating spray fluidized bed agglomeration of porous particles. The effect of micro-processes on aggregation kinetics is analyzed using a sophisticated constant number Monte-Carlo (CNMC) approach. The CNMC is used as a virtual spray fluidized bed granulator and takes into account several micro-processes such as wetting, convective drying, imbibition, agglomeration etc. so that it is a model for the real granulator. The main components influencing the agglomeration kinetics, namely the total number of particles, the number of wet particles, the collision frequency per particle and the average wet surface coverage fraction per wet particle are modeled and incorporated in the conventional one-dimensional population balance model. Finally, the proposed population balance model is validated against the results of CNMC.

Population balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granulation

In this work, one-dimensional population balance models (PBMs) have been developed to model a pulsed top-spray fluidized bed granulation. The developed PBMs have linked the key binder solution spray operating factors of the binder spray rate, atomizing air pressure and pulsed frequency of spray with the granule properties to predict granule growth behaviour in the pulsed spray fluidized bed granulation process at different operating conditions with accuracy. A multi-stage open optimal control strategy based on the developed PBMs was proposed to reduce the model mismatch, in which through adjusting the trajectory of the evolution of the granule size distribution at predefined sample intervals, to determine the optimal operating variables related to the binder spray including the spray rate of binding liquid, atomizing air pressure and pulsed frequency of spray. The effectiveness of the proposed modelling and multi-stage open optimal control strategies has been validated by experimental and simulation tests.

Analysis of breakage and coalescence models for bubble columns

This work was aimed at evaluating the performance of bubble breakage and coalescence models in bubble column simulations. A total of five different models were considered (two for breakage and three for coalescence). These selected models had their parameters estimated using experimental data of bubble size distributions for the air–water system in an isothermal bubble column. Bubble sizes were measured with a photographic technique for two gas superficial velocities at three bubbling heights. Model parameters were estimated using the maximum likelihood technique applied to a one-dimensional population balance model, which was solved by the method of classes. The results allowed to conclude, within the experimental uncertainty, that it was necessary to adjust the parameters of some of the models according to the operational conditions. For other models, the same parameter values could be used to represent different operational conditions.

Modeling of Particle Size Distribution for Semibatch Precipitation of Barium Sulfate Using Different Activity Coefficient Models

Our recent experiments with precipitation of barium sulfate (Steyer et al. In BIWIC 15th International Workshop on Industrial Crystallization; Shaker VerlagAachen, Germany, 2008; pp 73−80. Steyer and Sundmacher J. Cryst. Growth 2009, 311, 2702−2708) showed that an excess of barium or sulfate ions in the crystallizer has a strong impact on the resulting particle size distributions (PSDs). Theoretically, this effect is not well understood, as most kinetic models in the literature are valid only for stoichiometric conditions. The aim of this work was, therefore, to investigate the influence of detailed thermodynamically well formulated activity coefficient models on the PSDs in a one-dimensional population balance model of a semibatch continuous stirred-tank reactor (CSTR). The nonsymmetry of these models with respect to an excess of barium or sulfate ions was studied. It was found that the effect on the supersaturation is too weak to explain the experimental results in terms of supersaturation-dependent kin...

Model predictive control of continuous drum granulation

This paper details a methodology for the design of a model predictive controller for a continuous granulation plant. The work is based on a non-linear one-dimensional population balance model (1D-PBM), which was parameterized using experimental step test data generated at a continuous granulation pilot plant installed at the University of Queensland, Australia. The main objective was to operate the granulator under optimal conditions while off-specification material was fed back into the granulator to increase the economy of the process. The final algorithm design combines elements of model predictive control (MPC) with gain scheduling to cancel non-linearities in the recycle flow. A model directly identified from the step test data was the basis for testing a model predictive controller. Simulations show that the efficiency and robustness of this granulation process can be improved by applying the proposed control strategy. Ongoing work focuses on the implementation of the proposed control strategy on a full scale industrial plant.

Two state estimators for the barium sulfate precipitation in a semi-batch reactor

The on-line determination of particle property distributions by direct measurements is often difficult, because the measurement equations are not invertible or because the inverse problem is ill-posed. If the process is observable, one can use state estimation techniques in order to reconstruct unmeasurable internal states of the process. This is discussed here for a semi-batch precipitation reactor. A square root unscented Kalman filter and state estimation by online minimisation are studied for the case of a measurable average particle size. Both estimators use a one-dimensional population balance model. The two approaches are compared in simulations.

Analytical and numerical investigations of a batch crystallization model

This article is concerned with the analytical and numerical investigations of a one-dimensional population balance model for batch crystallization processes. We start with a one-dimensional batch crystallization model and prove the local existence and uniqueness of the solution of this model. For this purpose Laplace transformation is used as a basic tool. A semi-discrete high resolution finite volume scheme is proposed for the numerical solution of the current model. The issues of positivity (monotonicity), consistency, stability and convergence of the proposed scheme for the current model are analyzed and proved. Finally, we give a numerical test problem. The numerical results of the proposed high resolution scheme are compared with the solution of the reduced four-moments model and the first-order upwind scheme.

A comparative study of high resolution schemes for solving population balances in crystallization

This article demonstrates the applicability and usefulness of high resolution finite volume schemes for the solution of population balance equations (PBEs) in crystallization processes. The population balance equation is considered to be a statement of continuity. It tracks the change in particle size distribution as particles are born, die, grow or leave a given control volume. In the population balance models, the one independent variable represents the time, the other(s) are “property coordinate(s)”, e.g. the particle size in the present case. They typically describe the temporal evolution of the number density functions and have been used to model various processes. These include crystallization, polymerization, emulsion and cell dynamics. The high resolution schemes were originally developed for compressible fluid dynamics. The schemes resolve sharp peaks and shock discontinuities on coarse girds, as well as avoid numerical diffusion and numerical dispersion. The schemes are derived for general purposes and can be applied to any hyperbolic model. Here, we test the schemes on the one-dimensional population balance models with nucleation and growth. The article mainly concentrates on the re-derivation of a high resolution scheme of Koren (Koren, B. (1993). A robust upwind discretization method for advection, diffusion and source terms. In C. B. Vreugdenhill, & B. Koren (Eds.), Numerical methods for advection–diffusion problems, Braunschweig: Vieweg Verlag, pp. 117–138 [vol. 45 of notes on numerical fluid mechanics, chapter 5]) which is then compared with other high resolution finite volume schemes. The numerical test cases reported in this paper show clear advantages of high resolutions schemes for the solution of population balances.

Modelling nucleation in wet granulation

Nucleation is the first stage in any granulation process where binder liquid first comes into contact with the powder. This paper investigates the nucleation process where binder liquid is added to a fine powder with a spray nozzle. The dimensionless spray flux approach of Hapgood et al. (Powder Technol. 141 (2004) 20) is extended to account for nonuniform spray patterns and allow for overlap of nuclei granules rather than spray drops. A dimensionless nuclei distribution function which describes the effects of the design and operating parameters of the nucleation process (binder spray characteristics, the nucleation area ratio between droplets and nuclei and the powder bed velocity) on the fractional surface area coverage of nuclei on a moving powder bed is developed. From this starting point, a Monte Carlo nucleation model that simulates full nuclei size distributions as a function of the design and operating parameters that were implemented in the dimensionless nuclei distribution function is developed. The nucleation model was then used to investigate the effects of the design and operating parameters on the formed nuclei size distributions and to correlate these effects to changes of the dimensionless nuclei distribution function. Model simulations also showed that it is possible to predict nuclei size distributions beyond the drop controlled nucleation regime in Hapgood's nucleation regime map. Qualitative comparison of model simulations and experimental nucleation data showed similar shapes of the nuclei size distributions. In its current form, the nucleation model can replace the nucleation term in one-dimensional population balance models describing wet granulation processes. Implementation of more sophisticated nucleation kinetics can make the model applicable to multi-dimensional population balance models.

Limitations of one-dimensional population balance models of wet granulation processes

Most traditional models of wet granulation processes use one-dimensional population balances, which assume that granule size is the only independent granule property that significantly affects granule growth behaviour. However, several other independent granule properties have been identified, which can also strongly influence granulation behaviour. These include granule binder content, porosity, and primary particle size distribution and composition. Most population balance models also implicitly assume that conditions are spacially uniform throughout the granulator. However, segregation has the potential to occur in many commercial granulators. This will alter the frequency and velocity of collisions between different sizes of granule. Hence, current one-dimensional models of wet granulation are inadequate. These models need to be expanded to three or four dimensions to include size, porosity, binder content, and composition (where applicable) as independent granule properties and also need to include the effects of mixing and segregation within the granulator.