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© 2021 MJH Life Sciences^{™} and BioPharm International. All rights reserved.

April 1, 2010

Zhiwu (David) Fang##### Zhiwu (David) Fang

**BioPharm International**, BioPharm International-04-01-2010, Volume 23, Issue 4

Zhiwu (David) Fang is the founder and principal consultant of Systems Quality-by-Design, Inc.

*Computational fluid dynamics is a powerful tool to optimize processes.*

**The capabilities of computational fluid dynamics (CFD) as a tool for trouble-shooting and Quality by Design have been demonstrated through case studies in bioprocesses from upstream, to downstream, to fill–finish. The applications involve, for example, multiple physics, fluid dynamics, transport phenomenon, heat transfer, phase transition, and rheology. Flow types include laminar flow in emulsification, turbulent flow in bioreactor mixing, and porous media flow in chromatographic columns, filtration, and centrifuge machine. Part 1 of this article series is an overview of the technology, followed by a discussion of the model setup. Part 2 of this series will discuss how CFD is used to perform flow characterization, mixing, resuspension, fermentation, and cell culture.**

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows and associated phenomena. CFD is a highly sophisticated integration of applied computer science, physics, chemistry, and engineering science. Existing commercial CFD codes are capable of simulating a very wide variety of physical processes besides fluid flow. To date, there are more than a dozen commercially available codes (e.g., Ansys, Fluent, CFX, Star-CD, Flow3D, and Phoenics) and open-source codes (e.g., Open-Foam) for CFD modeling and simulations. Significant improvements are continually being made in both CFD software and the hardware used to run it. The software is becoming more user friendly and the hardware is providing more computing power for less money.

(AMGEN, INC.)

Besides being a powerful instrument to analyze "what if" scenarios to improve the efficiency of existing operating systems, CFD also can be used as a Quality by Design (QbD) tool for the design of new systems. It can help shorten product and process development cycles; optimize processes to improve robustness, efficiency, and productivity; and solve problems as they arise in unit operations. CFD technology is well established and has been used successfully in the aerospace and automotive industries in optimizing control strategies and equipment, and shortening the time from design to production. It entered the chemical industries over the past two decades and has been applied to various chemical processes.^{1,2} Recently, CFD has seen increased interest in the pharmaceutical arena, where insights into fluid flow and related phenomena can help mitigate risks associated with scaling-up process equipment and aid in troubleshooting.^{3,4} Furthermore, it has demonstrated its potential as a QbD tool.

Pharmaceutical and biopharmaceutical processes that can benefit from CFD analysis include turbulent flow in bioreactors; multiphase flow, heat, and mass transfer; chemical reactions; phase transitions; porous media flow; and granular particle material dynamics. These processes cover upstream, downstream, and fill–finish, and involve unit operations such as fermentation and cell culture, mixing, chromatography, filtration, centrifuge separation, product freezing and thawing, lyophilization, emulsification, spray drying, product packing, and transportation. With growing capabilities and flexibility, CFD has been the most widely applicable tool available to simulate the physical processes involved in process development, manufacturing, and drug delivery.

There have many reviews of the application of CFD to various industrial processes, only a few of which have covered the role of CFD in the pharmaceutical industry.^{3,5,6} The biotechnology industry has its own unique processes, in addition to shared common unit operation processes with the pharmaceutical industry. This review will focus on applying CFD to bioreactor (fementer) operations.

The fundamental basis of any CFD problem is the Navier-Stokes equations that describe the motion of fluids. Additional physiochemical processes are coupled with fluid flow, such as multiphase (gas–liquid, or solid–liquid, or solid–liquid–gas) interaction, species transport, heat transfer, mass diffusion, and chemical reactions. Numerical algorithms are used to solve conservation equations for mass, momentum, and energy, and provide solutions on flow and physical variables such as velocity, pressure, temperature, density, concentration, and volume or mass fraction, from which other derived variables such as shear stress, pH, height equivalent to a theoretical plate (HETP), etc., are obtained.

CFD involves two procedures of discretization—domain discretization and equation discretization—followed by the application of a suitable algorithm to solve the resulting ordinary differential equations for unsteady problems and an algebraic equation for steady problems, subject to appropriate boundary conditions and initial conditions, to get solutions on the grid.^{7} Either an iterative solver or a direct solver can be used, depending on the size of the problem, available computer memory, and CPU speed. In the step of equation discretization, partial differential equations are discretized into a finite difference equation, whereas in domain discretization, the continuous spatial domain is discretized into small control volumes to form a volume mesh or grid. A postprocessor is used for the analysis and visualization of the resulting solution.

Mesh-based methods include the finite difference method (FDM), the finite volume method (FVM), the finite element method (FEM), the boundary element method (BEM), and some high-resolution schemes that treat shocks and discontinuities. Non-mesh–based methods include smoothed particle hydrodynamics (SPH)—a Lagrangian method of solving fluid problems, Spectral methods—a technique where the equations are projected onto basis functions like the spherical harmonics and Chebyshev polynomials, and Lattice Boltzmann methods (LBM), which simulate an equivalent mesoscopic on a Cartesian grid instead of solving the macroscopic system (or the real microscopic physics).

The key to a successful CFD analysis of a bioprocess is to choose correct models based on underlying physics and biochemistry. These models also require the right boundary conditions and initial conditions.

Currently, there are two broadly categorized computational approaches for modeling the interaction between phases in multiphase flows: the Euler-Lagrange approach^{8–11} and the Euler-Euler approach.^{12–17} In the Euler-Lagrange approach, the fluid phase is treated as a continuum by solving the time-averaged Navier-Stokes equations, whereas the dispersed phase is solved by tracking a large number of particles through the calculated flow field. The Euler-Lagrange approach typically deals with the dispersed second phase by occupying a low volume fraction, even though high mass loading is acceptable. The particle trajectories are computed individually at specified intervals during the fluid phase calculation. In the Euler-Euler approach, the different phases are treated mathematically as interpenetrating continua. Because the volume of a phase cannot be occupied by the other phases, the concept of phasic volume fraction is introduced. These volume fractions are assumed to be continuous functions of space and time and their sum is equal to one. Conservation equations for each phase are derived to obtain a set of equations that have similar structures for all phases. In both approaches, the continuum phase and discrete phase momentum equations are coupled through the drag source and sink terms, and through the volume fraction of the dispersed phase.

The flow in an agitated bioreactor typically is in a fully turbulent regime, although some emulsification processes may operate in a laminar or transitional regime because of the high viscosity of the mixture. Two methods often used to model the turbulent flow are the Reynolds-averaged Navier-Stokes (RANS) models and the unsteady large eddy simulation (LES) models. The direct numerical simulation (DNS) method is too computationally demanding for real applications.

RANS-based models have different variations, such as the standard k-epsilon, the Chen-Kim, the renormalized group (RNG), the realizable k-epsilon, and the k-omega, which all assume isotropic turbulence and do not show superiority of one over another,^{18} and the Reynolds stress transport models (RSTM), which incorporate the anisotropy characteristics by solving transport equations for Reynolds stress terms. However, the improvement on the accuracy of RSTM predictions over the k–e models is not conclusive.^{19–24} Based on the Kolmogorov theory, the LES models solve only for the large eddies explicitly, while the small-scale effects, below the filter size with corresponding wave number lying in the inertial convective subrange of the energy spectrum, are modeled using a sub-grid scale model. Good agreement between experimental data and model predictions on mean velocities and turbulence quantities in agitated tanks have been reported.^{22,25–33} LES requires less computational effort than direct numerical simulation (DNS), which is the most exact approach to model turbulence and can resolve the smallest eddies, but uses significantly more effort than methods that solve RANS.^{34}

When modeling a rotating system such as in an agitated tank, the multiple reference frame (MRF) and the sliding mesh (SM) model often are used. The MRF model performs a steady-state calculation with a rotating reference frame in the impeller region and a stationary reference frame in the outer region. In this way, the effects of the impeller rotation are accounted for by the frame of reference, allowing for explicit modeling of the impeller geometry. The SM model allows the impeller region to slide relative to the outer region in discrete time-steps and performs time-dependent calculations using implicit or explicit interpolation of data at successive time-steps. The SM model is a more accurate representation of the actual phenomenon of the impeller rotation, but unfortunately a computationally demanding one.

The predictive capabilities of all available CFD models when applied to agitated bioreactors equipped with various impellers in up- or down-pumping mode are discussed in the reviews.^{23,24,36–38} In summary, all turbulent models can predict the mean flow-field and power number very well,^{23,30,31,39} and also can capture most of the key features of near-impeller flows with sufficient accuracy, but provide various degrees of agreement with experimental data on turbulent characteristics. The standard k-epsilon turbulence model combined with the MRF model, as commonly used in engineering CFD simulations of stirred tanks and often faulted for its assumption of isotropic turbulence, can model the turbulent flow with adequate accuracy if fine enough grids coupled with higher-order discretization schemes are used.^{38} The LES for modeling flow in stirred tanks has the advantage of capturing the instantaneous velocity field and vortex structures.^{22,25,28,30,31,40}

Cells growing in bioreactors take up nutrients from the culture medium and release products, byproducts, and waste metabolites. Mixing and sparging greatly influence the mass transfer phenomena required for an adequate supply of nutrients and removal of waste metabolites.^{35} Agitation is used to maintain cells in suspension, provide a homogeneous mix of nutrients, and prevent the accumulation of toxic gases. Multiple tasks and numerous choices of impellers make process scale-up extremely challenging. Figure 1 illustrates how CFD provides assistance in understanding the extracellular environment, optimizing the operation conditions, designing hardware configuration, and implementing scale-up and scale-down strategies.

Figure 1

Even though CFD has advanced remarkably, many challenging cases require CFD experts. Applying tools blindly without understanding the capabilities and limitations of the methods involved could lead to erroneous results. Besides good knowledge of numerical computation technologies (i.e., ensuring mesh quality and selecting time-steps for unsteady simulations), understanding the underlying physics, chemistry, and biology, and choosing the most appropriate physiochemical models are vital to successful CFD simulations. For example, a model taking interparticle and intraparticle mass transport and adsorption for chromatographic separations into account can be a powerful predictive tool. However, industrial process streams contain many proteins and an array of other materials such as lipids, nucleic acids, and cell debris, all of which may interact with the column matrix. Determining their competitive isotherms individually is not realistic. Simplified but reasonably accurate isotherm models based on theoretical study or through experimental data determination should be set up for any CFD modeling effort.

Making prediction with high confidence would require including not only fluid dynamic modeling but also modeling of other physical quantities. Areas that relate to bioprocesses that still need extensive research include interphase drag laws, bubble breakup, and coalescence mechanisms; constitutive models for the deformation of soft porous media; the dependence of cell damage on energy dissipation rate; cohesion; and multiple-particle interactions for spherical and nonspherical particles, just to name a few. With regard to the computation technologies, a substantial reduction in computational time for 3D unsteady flow simulations can further promote the application of CFD in the biopharmaceutical industry. CFD simulations produce huge amounts of raw data for unsteady flow problems, therefore, data storage and management will become a critical issue.

CFD model validation is necessary in any CFD modeling effort. Designing experimental verification requires the same level of understanding of the physiochemical mechanisms as for CFD model setup. Choosing parameters for comparison is a process of scientific reasoning, as well as artistic intuition.

When designing fermentation or cell culture bioreactors, one needs to address various issues. Among them are the simultaneous dispersion of gas, the homogenization (mixing) of the nutrients or base, the suspension of living cells by the impellers, and mass transfer between cells and media. Media and bubbles are of vital importance to the performance of a bioreactor. In the case of airlift bioreactors, air flowing upward in a column-shaped bioreactor vessel generates sufficient mixing of gases and cells simultaneously, thereby replacing the need for the conventional impellers of a stirred tank bioreactor. Aeration coupled with agitation in conventional bioreactors generates complex flow dynamics in the tank. Furthermore, foaming resulting from the high volume of airflow will adversely affect process performance. Bubble busting at the air–medium interface has been reported to cause cell damage. As indicated in Figure 2, considerations of simultaneously suspending cells (solids), dispersing gas completely, achieving a sufficient ratio of surface area to gas volume for mass transfer, and minimizing detrimental hydrodynamic forces leave a very small design and performance space. Finding the overlapped optimal space for each bioreactor and using it as the scale-up and scale-down and design criterion is a challenging but rewarding task.

Figure 2

The use of CFD has gone through many great developments, in terms of the computational technologies for robust, accurate, and efficient numerical analysis tools, as well as higher levels of sophistication of physical modeling in the areas such as turbulence and multiphase flow.

Although the semi-empirical correlations or the lack of sound physics principles in CFD models limit its predictive capabilities, undoubtedly, with current computing power progressing unrelentingly, multiscale modeling and simulations from the particle level to the continuum level will become more and more realistic and uncover more fundamental physics. It is conceivable that CFD will continue to provide explanations for more and more flow-related phenomena. Fueled by science-centered regulatory initiatives and cost and quality concerns, the use of CFD modeling technologies will provide significant opportunities for optimization and quality enhancement in the future.

The author would like to thank his former colleagues in process development, manufacturing and research at Amgen for explaining their needs to him and enriching his understanding of bioprocesses through many years of collaboration.

**Zhiwu (David) Fang** is the founder and principal consultant of Systems Quality-by-Design, Inc., Newbury Park, CA, 626.716.2016, zfang@systemsqbd.com

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