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Integrated Population Balance and Multiphysics Modeling for Predicting Undesired Agglomeration in Small Molecule Manufacturing
Prakitr Srisuma, Peter Hou, Shashank Venkat Muddu, Neda Nazemifard, Allan S. Myerson, Richard D. Braatz
TL;DR
Hard agglomerates irreversibly alter product particle size and can cause off-spec products, equipment damage, and downstream processing, while existing models do not distinguish their formation from soft agglomeration. The paper presents a mechanistic model integrating population balance and multiphysics modeling. It accurately predicts temperature, moisture, and particle size in two experimental systems and is used for model-based design and optimization.
Problem
Existing AFD models do not explicitly predict particle size distribution while distinguishing hard from soft agglomerates, despite hard agglomerates irreversibly changing product PSD and causing quality and equipment problems.
Source: §1 · pp. 1–2Method
The paper develops a mechanistic AFD model combining population balance and multiphysics modeling of heat transfer, mass transfer, particle interactions, agglomeration, and breakage.
Source: §1 · p. 2Results
The model accurately predicts product temperature, moisture content, and particle size, particularly D90 and D99, across two experimental systems with distinct agglomeration behaviors.
Source: §5 · pp. 13–14Takeaways & Limitations
The validated model provides a framework for system analysis and model-based design and optimization to minimize hard agglomerate formation.
Source: §5 · p. 14The framework's applications are limited to those considered in the study; integration with state estimation and model predictive control remains future work.
Source: §5 · p. 14
Abstract
from arXiv · showhide
Agitated filter dryers (AFDs) are a crucial unit operation in small molecule manufacturing that enables simultaneous filtration, washing, and drying of active pharmaceutical ingredients. One of the key challenges in AFDs is associated with undesired agglomeration, where the presence of hard agglomerates results in off-spec products, equipment damage, and additional downstream processing. This article presents a novel mechanistic model that describes the formation of soft and hard agglomerates during agitated filter drying. By integrating population balance and multiphysics modeling, the model can accurately predict the evolution of the product temperature, moisture content, and particle size distribution, and hence quantify the extend and impact of undesired agglomeration across various operating conditions. Our proposed model-based framework enables the rational design and operation of AFDs for improving the product quality and process reliability.
1 Introduction
Hard agglomeration is a key AFD challenge because it irreversibly changes product particle size and can cause quality and equipment problems. Existing models do not distinguish hard from soft agglomerate formation, motivating a mechanistic model that integrates population balance and multiphysics modeling.
1 Introduction
- AFDs combine filtration, washing, and drying to recover purified, dry API crystals, while agitation can promote particle agglomeration.Agitation can also enhance heat and mass transfer.Source: §1 · p. 1
- Hard agglomerates irreversibly change product PSD, causing off-spec products, equipment damage, and additional downstream processing.The paper identifies minimizing hard agglomerate formation as crucial in AFDs.Source: §1 · p. 1
- Experimental studies have examined operating conditions and material properties including solubility, agitation, drying temperature and pressure, solvent content, and particle morphology.These studies investigate their influence on particle agglomeration during agitated filter drying.Source: §1 · p. 1
- Existing models do not predict PSD while explicitly distinguishing hard and soft agglomerates, leaving undesired agglomeration incompletely described.Prior approaches include heat- and mass-transfer models, DEM, and population balance models.Source: §1 · p. 2
- The proposed mechanistic model combines population balance and multiphysics modeling to describe soft and hard agglomerate formation in AFDs.It is evaluated on two experimental systems representing absent and present hard agglomerates.Source: §1 · p. 2
2 Experimental Setup and Process Description
The experimental system uses an agitated filter dryer with inline monitoring of process conditions and particle size. Sampling during drying provides moisture measurements over time.
2 Experimental Setup and Process Description
- The AFD setup includes a dryer, jacketed filter basket, agitator, temperature and pressure sensors, and a sight glass for laser-speckle PSD monitoring.Powder is soaked in solvent while agitated at 5 rpm for 15 minutes, then filtered before the automated drying sequence.Source: §2 · p. 2, Figure 1 · p. 3
- Inline instrumentation records powder-bed surface temperature and vacuum pressure during drying.A non-contact infrared sensor measures temperature, and the HMI records the profiles.Source: §2 · p. 2
- Laser-speckle images are converted into transient PSD profiles using the PEACE framework, and drying ends when mean particle size stabilizes within 5% for at least 5 minutes.The optical setup continuously collects backscattered light from the moving powder surface.Source: §2 · p. 3
- Moisture profiles are measured by pausing drying every 30 minutes to collect and gravimetrically analyze samples.Samples are weighed, dried overnight under vacuum, and reweighed to calculate moisture content.Source: §2 · p. 3
3 Mechanistic Modeling
The mechanistic framework couples heat and mass transfer with population-balance descriptions of agglomeration and breakage, distinguishing soft and hard agglomerates during drying.
3 Mechanistic Modeling
- The model represents heat transfer, mass transfer, and particle agglomeration and breakage as the three main aspects of agitated filter drying.Source: §3 · p. 3
3.1 Heat transfer
- Assuming a well-mixed system, the heat-transfer model accounts for heating fluid, inert gas, agitation work, and evaporative cooling.Source: §3.1 · p. 3
3.2 Mass transfer
- Drying is modeled in constant-rate and falling-rate regimes: evaporation is governed by external transfer in the first and internal solvent diffusion in the second.Source: §3.2 · pp. 4–5
3.3.1 Population balance modeling
- A population balance tracks the evolving particle-size distribution as particles agglomerate into larger particles or break into smaller ones.Source: §3.3 · p. 5, §3.3.1 · p. 5
3.3.2 Formation of soft and hard agglomerates
- Soft agglomerates can return to primary particles, while hard agglomerates form when crystallization creates solid bridges before evaporation removes liquid bridges.Source: §3.3.2 · p. 6
- The soft-to-hard conversion rate is linked to liquid evaporation and solute crystal growth, whose rate is expressed volumetrically for comparison with evaporation.Source: §3.3.2 · pp. 6–7
3.3.3 Agglomeration and breakage kernels
- Agglomeration and breakage kernels depend on moisture content, with agglomeration favored at higher moisture and deagglomeration at lower moisture.Source: §3.3.3 · p. 7, Table 1 · p. 8
3.3.4 Breakage probability function
- The breakage probability distribution is assumed to follow the initial particle-size distribution because soft agglomerates tend to return to primary particles.Source: §3.3.4 · p. 8
3.4 Gas dynamics
The gas-dynamics extension is optional when operating pressure changes little over time. The final model tracks product temperature, moisture, and soft- and hard-agglomerate number densities, with gas temperature and pressure included when pressure varies significantly.
3.4 Gas dynamics
- The gas-dynamics component is optional and is intended for cases where operating pressure changes significantly over time.AFD pressure can be varied by changing the amount of inert gas, which can influence heat and mass transfer.Source: §3.4 · p. 8
3.5 Model summary and numerical methods
- When pressure varies significantly, the model can track operating pressure and inert-gas temperature alongside four core states: product temperature, moisture, and soft- and hard-agglomerate number densities.The gas-temperature calculation assumes adiabatic expansion and compression of an ideal gas; specified pressure profiles determine gas temperature.Source: §3.5 · p. 8, §3.4 · p. 8
- The model summarizes nonlinear functions derived from energy, mass, and population balances, and discretizes the population-balance PDEs by finite volumes before ODE integration.The temperature and moisture equations can be integrated directly with an ODE solver; particle-size bins index the discretized system.Source: §3.5 · pp. 8–9, Figure 4 · p. 9
4 Results and Discussion
The model was validated against two AFD systems, predicting temperature, moisture, and particle-size behavior in a soft-agglomerate-only case and a case where hard agglomerates form. Simulations then distinguish reversible soft-agglomerate behavior from persistent hard agglomerates and examine how solubility and mass transfer relate to final D99.
4.1 Model validation
- In KCl–hexane, predicted temperature, moisture, and D90/D99 agree with experiments; D90 and D99 return to initial values as soft agglomerates deagglomerate.Complete KCl insolubility in hexane prevents hard-agglomerate formation; temperature rises toward the hot-oil temperature while residual moisture falls toward its target.Source: §4.1 · pp. 9–10, Figure 5 · p. 10
- In ASA–IPA, the model predicts higher final D90 and D99 than initial values and the final particle-size distribution, consistent with hard-agglomerate formation.The simulated temperature and moisture trends closely align with experiments; ASA’s high solubility in IPA promotes hard-agglomerate formation.Source: §4.1 · pp. 10–11, Figure 6 · p. 11
- Across KCl–hexane and ASA–IPA, the model predicts temperature, moisture, and particle-size behavior, including D90 and D99, for cases with and without hard agglomerate formation.The authors report physically reasonable predictions across different mixtures and operating conditions.Source: §4.1 · pp. 9–11
4.2 Model-based analysis, design, and optimization
- Without hard agglomerates, the simulated final PSD and D90/D99 return to their initial values, particle count recovers, and total particle mass remains conserved.The PSD first shifts toward larger sizes as agglomeration dominates, then shifts back as deagglomeration becomes dominant.Source: §4.2 · p. 11, Figure 7 · p. 12
- With hard agglomerates, the final PSD differs from its initial distribution, D90/D99 remain higher, and particle count does not recover, while total mass is conserved.The simulated final PSD shows a slight mismatch from the initial PSD at 217 min.Source: §4.2 · pp. 11–12, Figure 8 · p. 13
- Higher solubility promotes hard agglomeration and larger D99, whereas higher mass-transfer rates reduce solid-bridge formation probability and lower D99.The model quantitatively captures these competing effects to inform AFD design and operation aimed at mitigating undesired agglomeration.Source: §4.2 · pp. 12–13
5 Conclusion
The paper presents a mechanistic model integrating heat transfer, mass transfer, and population balances to describe soft and hard agglomeration during agitated filter drying. It validates the model against two experimental cases and proposes extending the framework's applications.
5 Conclusion
- The model was validated against KCl–hexane data with only soft agglomerates and ASA–IPA data with significant hard agglomerates, predicting temperature, moisture, and particle-size evolution.Source: §5 · p. 13
- D90 and D99 are identified as especially relevant to hard-agglomerate formation, and parametric studies demonstrate model-based design and optimization to minimize hard agglomerates.Figure 9 presents end-of-process D99 values as a three-dimensional surface.Source: §5 · p. 14, Figure 9 · p. 14
- Future work could extend the framework beyond the applications considered, including integration with state estimation and model predictive control for real-time monitoring and particle-size control.Source: §5 · p. 14
A Model and Code Verification
The section performs model and code verification because numerical implementation of population balances with simultaneous agglomeration and breakage is nontrivial.
A Model and Code Verification
- The section verifies the model equations and numerical scheme because implementing simultaneous agglomeration and breakage in population balances is nontrivial.Source: A Model and Code Verification · p. 14
A.1 Comparison with analytical solutions
For two simplified cases, the numerical solutions closely match analytical solutions across particle size and time.
A.1 Comparison with analytical solutions
- Numerical solutions are nearly identical to analytical solutions across size and time for exponential-like Case A and Gaussian-like Case B initial particle-size distributions.Source: §A.1 · p. 14, Figure A.1 · p. 15
A.2 Error analysis
The error analysis finds that numerical-versus-analytical solution errors decrease as the number of bins increases.
A.2 Error analysis
- RMSE errors between numerical and analytical solutions decrease as the number of bins increases, as expected for a proper discretization scheme.Figure A.2 shows the errors for Cases A and B measured at t = 100 s.Source: §A.2 · p. 15, Figure A.2 · p. 15
A.3 Physical constraint satisfaction
The numerical solutions conserve total particle mass exactly in Cases A and B. The appendix reports agreement with analytical solutions and reasonable error analysis.
A.3 Physical constraint satisfaction
- Total particle mass is conserved exactly in Cases A and B (Fig. A.3).The figure tracks total mass over time for both cases.Source: §A.3 · p. 15, Figure A.3 · p. 15
- Any deviation from mass conservation, even a small one, indicates an implementation error in the population balance model.Mass conservation must hold throughout agglomeration and breakage.Source: §A.3 · p. 15
- The appendix reports agreement with analytical solutions, exact satisfaction of the physical constraint, and reasonable error analysis.Source: §A.3 · p. 16