A Review of Computational Fluid Dynamics (CFD) Simulations of Mixing in the Pharmaceutical Industry

mixing vessels. The pharmaceutical industry is not keen on publishing their research and the details of their technology, so the available literature is limited. In this mini review, publications, in which CFD simulations of the mixing process were performed for real pharmaceutical cases are presented, the obtained results are discussed and the software, modeling techniques and methods and their comparisons to measurements are highlighted.


Introduction
A literature review has shown that computational fluid dynamics (CFD) simulations in the pharmaceutical industry can be divided into the following topics: drug delivery and uptake, respiratory drug delivery, drying, and mixing, which is the topic of this review.
CFD simulations have become a valuable tool for process design, optimization, and scale-up [1]. In the pharmaceutical industry, stirred-tank reactors are used due to efficient mixing and the possibility of simple scale-up and appropriate mixing in agitated vessels directly relates to the quality of the product for both drug substance, specifically cell culturing and protein purification, and drug product manufacturing [2]. Simulations offer the possibility to obtain a deeper understanding of the evaluated process and can provide information that cannot be easily measured. Furthermore, CFD simulations consider the complete three-dimensional geometry as opposed to more basic correlations and various operational conditions and vessel/mixer geometries can be tested without the need for laboratory and large-scale operation. The presented CFD modeling papers include simulations made in collaboration with or supported by Roche [3], Sanofi [2], Sandoz [4], Krka d.d. [5], Pfizer Inc. [1,6], LEO Pharma A/S [7,8], Astra Zeneca Pharmaceutical Development R&D [9], GlaxoSmithKline [10,11] and one performed with the support of an industrial consortium including BASF, ICI, Malvern Instruments, AstraZeneca, GlaxoSmithKline and Pfizer [12].

Pharmaceutical Mixing Simulations Bioreactors
Ladner et al. [3] performed k-ε Reynolds-averaged Navier-Stokes (RANS) simulations with Flow 3D software to obtain shear rates in a stirred vessel, since they cannot be easily measured at specific DOI: 10.26717/BJSTR.2020.27.004494 20733 locations. Shear stress can damage sensitive microorganisms and the cell culture and consequently cause a decrease in productivity and can also influence the active pharmaceutical ingredient (API).
Bottom-mounted magnetic stirrers are commonly used in drug product manufacturing, because it is considered that they provide gentle mixing conditions. Magnetic stirrers have a thin gap between the stirrer head and the spigot through which the liquid also flows.
They compared the results to experiments with conductivity measurements. They found that although the shear rates in the gap were higher, the high shear rates in the spigot gap can be neglected due to the insignificant flow in that area. They also found that the direction of flow is dependent on the vessel volume, which should carefully be considered when designing small-scale (scale-down) models. Low shear inducing impellers are also used in bioreactors for cell production.
For providing low-shear mixing, the impeller blades usually have a low angle regarding the horizontal position, lower mixing speeds are used, and in the industrial production typically 2 impeller heads can be used for efficient mixing of the larger volume. Ebrahimi et al. [2] analyzed the effects of this impeller configuration and rotational speed on the mixing performance of a double-impeller bioreactor. As expected, an increase in the mixing speed increased total power consumption and cell stress, while the mixing time was decreased. The CFD model-obtained power numbers were compared to experimentally determined ones from torque measurements on the impeller and a very close agreement was found, which validated their modeling approach, using ANSYS Fluent software and k-ε RANS turbulent equations The typical bioreactor and the CFD solution presenting velocity contours and vectors is presented on Figure 1. In another work, a stirred pilotscale bioreactor was studied by Bach et al. [8] for Trichoderma reesei fermentations with the standard k-ε RANS model in ANSYS CFX.
The mixing time was determined with tracer experiments in the fermentation broth. The method of using CFD to predict the mass transfer coefficient was validated with experiments and it was also proven that is was as accurate as empirical correlations.
Eulerian-Eulerian multiphase flow simulations with the volume-offluid method for capturing the gas-liquid interface were performed by Haque et al. [12] with the standard k-ε turbulent model, shearstress transport (SST) model and the differential Reynolds-stress   Experiments were done at the laboratory scale and it was found that the type of agglomerate formed correlated with the level of agitation. It was presumed that the obtained agglomerate type was dependent on the differences in the collisions of primary crystals due to differences in the local degree of agitation. They found a correlation between the turbulence dissipation rate (ε) of crystals and agglomeration rate into a specific morphology. A flaky, loose agglomerate was produced by conditions with lower ε. Turbulence dissipation rate was therefore used as the scaling parameter for pilot plant and commercial scale crystallization operating conditions. For the modeling, they used the Realizable k-ε turbulence model in Fluent. In another work, pharmaceutical antisolvent crystallization of aspirin from ethanol (solvent) and water (antisolvent) was studied by Öner et al. [7]. CFD simulations were used to obtain the mixing characteristics at different agitator speeds. The changes of solvent concentration, density, and compartmental volume were taken into account by the model while assuming a dynamic flow between compartments during filling. ANSYS CFX software was used with the standard k-ε turbulence model. A study of a batch crystallizer was also done by Chew et al. [11] in Fluent, who studied paracetamol crystallization using a realizable k-ε turbulence model and large eddy simulation. They compared the operation of a conventional impeller driven batch crystallizer and an oscillatory baffled batch crystallizer and found that the particles precipitated in the latter case were of significantly higher quality.
Simulations have shown that the shear rate in the latter case was much higher and responsible for a higher nucleation rate, which provided particles of a smaller size (Figure 2).

Flow of Particles
Particulate flow and mixing behavior in the blending of dry powders were modeled and simulated by an Eulerian-Eulerian multiphase framework by Nguyen et al. [9] in Fluent. The equations of dense particulate flow in a mixer with a high shear rate were  For mixing simulations, the rotating frame of reference technique (or multiple reference frame) is also commonly applied for similar reasons of low computational cost and proven accuracy, as opposed to the more realistic simulation of the mixer rotation with moving (sliding) mesh methods, which require considerably longer computational times. The lattice Boltzmann method has been shown to be efficient for large-volume two-phase flow simulations, which are harder to simulate by the Navier-Stokes equations.

Non-Newtonian Fluids
The majority of work was performed in ANSYS Fluent (previously only Fluent) or ANSYS CFX with the standard, Realizable or RNG k-ε models. For modeling particle dissolution and crystallization, population balance modeling is the most frequently applied.