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Use of Computational Fluid Dynamics as a Tool for Establishing Process DesignSpace for Mixing in a Bioreactor

A. S. Rathore, C. Sharma, and A. PersadDept. of Chemical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi, India

DOI 10.1002/btpr.745Published online November 14, 2011 in Wiley Online Library (wileyonlinelibrary.com).

The concept of design space plays an integral part in implementation of quality bydesign for pharmaceutical products. ICH Q8 defines design space as the multidimensionalcombination and interaction of input variables (e.g., material attributes) and process param-eters that have been demonstrated to provide assurance of quality. Working within thedesign space is not considered as a change. Movement out of the design space is consideredto be a change and would normally initiate a regulatory post-approval change process.

Design space is proposed by the applicant and is subject to regulatory assessment and ap-proval. Computational fluid dynamics (CFD) is increasingly being used as a tool for mod-eling of hydrodynamics and mass transfer. In this study, a laboratory-scale aeratedbioreactor is modeled using CFD. Eulerian-Eulerian multiphase model is used along with

dispersed ke turbulent model. Population balance model is incorporated to account forbubble breakage and coalescence. Multiple reference frame model is used for the rotatingregion. We demonstrate the usefulness of CFD modeling for evaluating the effects of typical

process parameters like impeller speed, gas flow rate, and liquid height on the mass transfercoefficient (kLa). Design of experiments is utilized to establish a design space for the abovementioned parameters for a given permissible range of kLa. VVC 2011 American Institute ofChemical Engineers Biotechnol. Prog., 28: 382391, 2012

Keywords: computational fluid dynamics, mixing, bioreactor, stirred tank, mass-transfercoefficient, agitation, design space, QbD, design of experiments

Introduction

Aerated stirred reactors are the most common type of bio-reactors used for performing microbial fermentation or mam-

malian cell culture unit operations for production of

biological therapeutics like vaccines, hormones, proteins, and

antibodies.1 Optimum mixing in bioreactors is conducive for

efficient transfer of oxygen, for removal of carbon dioxide,

and for maintaining uniformity of concentrations of the ana-

lytes and the cell mass. However, excess mixing leads to

high shear stress and may cause cell damage, in particular

with shear sensitive cells such as the Chinese Hamster Ovary

(CHO) cells that are commonly used in mammalian cell cul-

tures. Maldistribution with respect to energy dissipation,

bubble size, and oxygen within the bioreactor has been

shown to significantly impact both the mass transfer within

the bioreactor and the cell damage.26

Bioreactors have increasingly been studied using computa-

tional fluid dynamics (CFD) for modeling of local hydrody-

namic conditions including fluid flow, gas-fluid mixing,

agitation, and mass transfer inside the bioreactor. CFD not

only helps in understanding these conditions at a micro scale

but also equips us to evaluate the effect of any of the opera-

tional process parameters on the hydrodynamics of the bio-

reactor and thereby optimization of the process.7 CFD has

been used as a tool to assist with scaling-up of bioreactors

such that the shear environment can be maintained similarvia control of the process parameters.8,9 CFD applications

also include improvement of bioreactor design and character-

ization with respect to tissue growth models, scaffolds, and

matrices.10 CFD models in these applications have been used

for estimation of the various output parameters of interest

including the shear stress, oxygen concentration, Sauter-

mean diameter, and mass-transfer coefficient.11

Quality by design (QbD) started gaining momentum in the

biotechnology industry after publication of the FDAs PAT

A Framework for Innovative Pharmaceutical Manufacturing

and Quality Assurance.12 Global acceptance of these princi-

ples is reflected in the contents of the International Confer-

ence on Harmonization (ICH) guidelines: ICH Q8

Pharmaceutical Development,13 ICH Q9 Quality Risk Man-

agement,14 and ICH Q10 Pharmaceutical Quality System.15 A

lot of recent publications have attempted to elucidate a path

forward for implementation of QbD and resolving the various

issues that otherwise serve as detriments to successful imple-

mentation.1618 QbD framework gives emphasis to identifying

critical process parameters and then prescribing a design

space to ensure a consistent product quality.19

In this article, we wish to demonstrate an approach for

establishing design space for mixing in a bioreactor utilizing

CFD simulations and the principles of design of experiments

(DOE).Correspondence concerning this article should be addressed to A. S.

Rathore at [email protected].

382 VVC 2011 American Institute of Chemical Engineers


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Theory

Quality by design

Quality by design is defined in the ICH Q8 guideline as

a systematic approach to development that begins with pre-

defined objectives and emphasizes product and process

understanding and process control, based on sound science

and quality risk management.13,16 Key steps of QbD imple-

mentation are identification of the product attributes that areof significant importance to the products safety and/or effi-

cacy (target product profile and critical quality attributes);

design of the process to deliver these attributes; a robust

control strategy to ensure consistent process performance;

validation and filing of the process demonstrating the effec-

tiveness of the control strategy; and finally, ongoing monitor-

ing to ensure robust process performance over the life cycle

of the product.1618 Furthermore, risk assessment and man-

agement, raw material management, use of statistical

approaches, and process analytical technology (PAT) pro-

vides a foundation to these activities.

Defining process design space

The concept of design space plays a key role in imple-

mentation of QbD for biopharmaceutical products. ICH Q8

defines design space as The multidimensional combinationand interaction of input variables (e.g., material attributes)and process parameters that have been demonstrated to pro-vide assurance of quality. Working within the design spaceis not considered as a change. Movement out of the designspace is considered to be a change and would normally initi-ate a regulatory post-approval change process. Design spaceis proposed by the applicant and is subject to regulatoryassessment and approval.13 The concept of process designspace is well understood concept in the biotechnology indus-

try.18,19 Key steps include performing risk analysis to iden-

tify parameters for process characterization; design studies

using DOE and perform using qualified scale-down models;

and finally, execute the studies and analyze the results to

define the design space. It should be emphasized that a com-

bination of acceptable ranges based on univariate experimen-

tation do not constitute a design space and for achieving the

latter the acceptable ranges must be based on multivariate

experimentation that take into account the main effects as

well as interactions of the process variables. This article

focuses on use of CFD in establishing process design space.

Design of experiments

DOE is a structured and organized method for experimen-tally determining the relationships between outputs of the

process (also called as responses) and the inputs of the pro-

cess (also called as factors). The experiments are designed

such that all the factors are systematically varied. The objec-

tive of such a study could be one or more of the following:

identification of optimal conditions, identification of factors

that have significant impact on the responses vs. those that

do not, and finally understanding of interactions between the

different factors. DOE can perform all these three tasks

more efficiently than the traditional one factor a time experi-

mental approaches. The output of a DOE study is a statisti-

cal model in the form of a mathematical equation that

predicts a given response variable as a function of the fac-

tors. Since a typical biotechnology unit operation consists of

1020 unit operation in series with 515 input parameters

and 510 output parameters, quality of a biotechnology prod-

uct can be impacted by 30100 parameters and their interac-

tions. This article uses a combination of DOE and CFD to

define mixing design space for a fermenter.

EulerianEulerian multiphase model

In this model, equations for conservation of mass and mo-mentum are derived for both liquid and gas phases and are

solved simultaneously. EulerianEulerian multiphase model

involves solving the Navier-Stokes equations assuming con-

stant density (q) and viscosity (l) of both phases. Mass andmomentum conservation equations for each phase, i, aregiven in volume-averaged form as follows:

@

@tqiai r:aiqiUi 0 (1)

@

@tqiaiUi r:qiaiUiUi airp r:sefRi Fi aiqig

(2)

where, qi, ai, and Ui represent the density, volume fraction,and mean velocity of phase i (liquid or gas), respectively.Pressure is denoted by p with terms Ri and Fi representingmomentum exchange and centrifugal forces in the rotating

frame. Further, sef is the Reynolds stress tensor and g is theacceleration due to gravity. Sum of the volume fractions of

both phases remains unity in each cell domain:

aL aG 1 (3)

Drag model

The drag force caused by the relative motion between the

two phases acts on the gas bubbles.Drag coefficient (CD) is calculated using Schiller-Nau-

mann correlation20 as a function of Reynolds number (Rep).

The correlation is as follows:

CD 24 1 0:15Re0:687p

Rep

Rep1000 (4)

CD 0:44 Rep > 1000 (5)

This model is applicable to bubbles in a still liquid. For tur-

bulent liquid, a modified law is used and the expression for

the Reynolds number has a modified viscosity term as shown

below21:

Rep qLUG ULd

lL Clt;L(6)

Where, C is the model parameter that accounts for the effectof turbulence in reducing slip velocity and has been assigned

a value of 0.3.22

Turbulence model

It is assumed that concentration of gas in the continuous

liquid phase is low (verified later from results of the model).

Hence, the dispersed ke turbulence model is used. The liq-

uid phase viscosity is given as:

Biotechnol. Prog., 2012, Vol. 28, No. 2 383


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lt;L qLClk2Liq

eL(7)

The turbulent kinetic energy (k) and the energy dissipationrate (e) are calculated from the respective equations of con-servation:

@qLaLkLiq

@t r:aLqLULkLip r: aL

lt;L

rk rkLiq

aLGkL aLqLeL aLqLPkL 8

@qLaLeL

@tr:aLqLULeL r: aL

lt;L

rereL

aLeL

kLiqC1eGkL C2eqLeL aLqLPeL 9

where GkL is the rate of production of turbulent kineticenergy and PkL and PeL account for the influence of dis-

persed phase on the continuous phase.23 Further Cl, C1e, C2e,rk, and re are model parameters having values equal to 0.09,

1.44, 1.92, 1.0, and 1.3, respectively.24

Population balance model

Single bubble size models are often used as they require

lesser computational time.25,26 While such models are simpler,

the real situation in the bioreactor is quite different. While the

bubbles are of a uniform diameter coming out of the sparger

hole, they undergo breakup and coalescence due to movement

of the primary liquid phase. Thus, instead of a single bubble

size, a wide spectrum of bubble sizes and shapes exists.

Breakup occurs when the surface tension of the bubbles is

overcome by the disruptive forces within the liquid. Coales-

cence occurs when the collisions between bubbles are strong

enough to break the liquid thin film. Models using population

balance equations (PBE) require higher amount of computa-

tion but do provide more accurate information on the second-

ary phase.7,27,28 There are several methods used for solving

population balance equations (PBE) such as method of classes

(discrete method) and the standard method of moments and

quadrature method of moments.29 In our study, method of

classes29,30 is used for solving population balance equations

for which the population balance equation can be written as:

qqGni

@tr:qGGni qGBiC DiC BiB DiB (10)

where ni is the number of particles (bubbles) in the ith bub-

ble class. BiB and BiC are the birth rates due to breakage andcoalescence, respectively, and DiB and DiC are the corre-sponding death rates. The breakage and coalescence terms

are modeled as functions of bubble volumes,29 V:

BiC 1

2

ZV0

aV V0; V0nV V0; tnV0; tdV0 (11)

DiC

Za0

aV; V0nV; tnV0; tdV0 (12)

BiB Za

V

mV0bV0pV; V0nV0; tdV0 (13)

DiB bVnV; t (14)

Where, a(V,V0) is the coalescence rate between bubbles ofsize V and V0, b(V0) is the breakage rate of bubble with sizeV, m(V0) denotes the number of daughter bubbles formeddue to breakage of size V0, n(V, t) denotes the number ofbubbles of volume V at time t and p(V, V0) is the probabilitydensity function for bubbles of size V, generated from bub-

ble of size V0.

Volume fraction of bubble size, i, is defined as:

ai niVi (15)

A new variable (fi) is defined as the ratio of the volume frac-tion of the i group bubbles and the total gas volume fraction:

fi ai

aG(16)

such that

Xifi 1 (17)

This term is directly used as a specific value boundary con-

dition when discrete method is applied while using FLU-

ENT. Equation 10 can now be written in terms of the Eqs.

15 and 16 as:

@qGfiaG

@tr:qGUGaGfi qGViBiC DiC BiB DiB

(18)

To couple population balance model with the fluid dynamics,

Sauter mean diameter (d32) is used as the input bubble diam-eter29 and is defined as:

d32 P

i

nid3

iPi nid

2i

(19)

Bubble breakage and coalescence model

The breakup model proposed by Luo and Svendsen is used

in this study.31 Isotropic turbulence in stirred tanks and binary

breakage having a stochastic breakage volume fraction has

been assumed. Breakage of bubbles occurs when turbulent

eddies with energy higher than bubble surface energy hit the

bubbles. Only eddies of length scale smaller or comparable

with the bubble size break the bubbles while larger eddies

just convect the bubbles. This model gives the breakage rate

of bubble size, Vj, into two bubbles of sizes, Vi and Vj Vi:

XBVj; Vi C2

d2j

12Z1nmin

1 n2

n113

exp 12cfr

b2qL223d

53

jn113

0@

1Adn(20)

where Cf is the increase in surface area due to breakage. k isthe arriving eddy size and k is the dimensionless eddy sizegiven by:

n k

dj(21)

The coalescence rate is written as a function of collision rate

(yij) and coalescence frequency (PC)32:

384 Biotechnol. Prog., 2012, Vol. 28, No. 2


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aVi; Vj hijPC (22)

where yij for bubbles per unit volume is given by:

hijp

4ninjdi dj

2e

1sd

23

i d23

j12 (23)

where d and n stand for bubble diameters and number den-

sities of two classes i and j. In general, coalescence can becaused by collision due to turbulence, buoyancy, and laminarshear.33 In our model, turbulence is taken as the dominant

phenomena that causes coalescence and the other two factors

are neglected.

Bioreactor Specifications and CFD Modeling

The bioreactor under consideration is a 3 L Brunswick

BioFlo 110 reactor with 2 L working volume and a tank

diameter of 12.5 cm.34 The impeller is attached to a 1 cm

central shaft and has three blades pitched at 45 degrees. The

air sparger is present as a 4 cm ring with six holes of 0.1-cm

diameter, each located 5 cm under the impeller. Earlier simula-

tions were done with a constant gas flow rate of 105 m3 s1

and an impeller rotation speed of 600 rpm. Energy equation

was not considered and material properties were taken at a

temperature of 20C. Properties of water used are as follows:

qL 998.2 kg m3, lL 0.001 kg m

1 s1, and rL

0.0719 N m1. Properties of air used are as follows: qG 1.225 kg m

3 and lG 1.7894 105 kg m

1 s1.

Geometry and mesh generation of the solution domain is

done using GAMBIT 2.4.6. An unstructured grid is created

containing 843,008 hybrid cells consisting of tetrahedral,

hexahedral, and wedge types. Figure 1 shows the different

parts of the bioreactor and the basic features of the grid.

Mesh quality was analyzed using skewness criteria. Skew-

ness value (equisize skew) between 0.50 and 0.80 is consid-ered acceptable and that below 0.50 is considered good.35 In

our case, 86% of cells had equisize value below 0.50 and

only 0.74% had value between 0.70 and 0.80. Aspect ratio

of 99% of the cells was between 1 and 2. Maximum cell

skewness and aspect ratio were found to be 0.78 and 2.2,

respectively. Thus mesh quality was acceptable. Initially, a

coarse mesh was created and the number of cells was

increased till the changes in the results were insignificant

(results on grid independency are presented later in the arti-

cle). FLUENT 6.3.26 was used as the solver code using the

finite volume method to discretize various governing equa-

tions into algebraic equations. Bubble diameter at sparger

hole is assumed to be uniform and is calculated using the

following correlation33,36

:

db 6rdh

gqL qG

13

(24)

where dh is the sparger hole diameter. Diameter found usingthe above expression was d 3 mm and was used for allthe initial simulations performed without PBE.

Multiple reference frames (MRF), dynamic mesh, and

sliding mesh are some of the approaches that have also been

used in the literature for CFD modeling.24 In the MRF

model, equations for the rotating region are solved using a

rotating frame of reference and the momentum conservation

equation (Eq. 2) has a centrifugal force term. The outside

region is solved using a stationary frame. In the Sliding

Mesh approach, the inner region actually rotates during sim-

ulation and slides along the interface with outer region and

the governing equations are solved in stationary frame for

absolute quantities. This method is more accurate than MRF

model, but requires much more computational time and

power which is why we used the MRF model for modelingthe impeller region (Figure 1). At the interface of inner and

outer region, continuity of absolute velocity is maintained to

match the solution. Furthermore, the upper boundary is con-

stituted as a gas outlet while applying a no-slip boundary

condition at the walls.

Phase-coupled SIMPLE algorithm is used for pressure-ve-

locity coupling. First-order Upwind and second-order Upwind

discretization schemes are used for solving the turbulence and

momentum equations, respectively. Volume fraction is discre-

tized using QUICK scheme as per consultation with Ansys

(manufacturer of Fluent software). This scheme is generally

used to discretize volume fraction for rotating or swirling

flows as it is more accurate than second order Upwind

scheme. For all other equations, second order scheme is suffi-cient. Further, the upper wall is constituted as a gas outlet by

applying the no-slip boundary condition at the wall. Solution

is assumed to be converged when (a) the scaled residuals of

all variables are smaller than 104, (b) the rate of gas leaving

and gas entering is equal, and (c) the values of various cus-

tom field functions used assume constant values.

Results and Discussions

Measurement of mass transfer coefficient and establishing

grid independency

Mass transfer in a bioreactor is often quantified via the

overall volumetric mass transfer coefficient (kLa). In this

Figure 1. Illustration of the geometry of the bioreactor underconsideration, the mesh used and the computationaldomain.



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article, Higbies penetration model is used for the calculation

of kLa. According to this model, a series of short chanceencounters take place between the two phases at the inter-

face resulting in unsteady molecular diffusion. Then, kLa isobtained from the product of local liquid mass transfer coef-

ficient (kL) and the interfacial area available for mass trans-fer (a). The equations are as follows:

kL 2

p

ffiffiffiffiffiffiffiffiDO2

p eqLlL

14

(25)

a 6aG

d32(26)

where the diffusion coefficient of oxygen, DO2, is taken as1.97 109 m2 s1.27 These equations are written as customfield functions in FLUENT to predict local and average kLain the whole computational domain.

Simulations were initially done with a coarse mesh for a

single bubble diameter throughout the domain. The number

of cells was increased to test for grid independency. Figure 2

shows that at a total cell count of 0.85 million cells results,

kLa, is independent of the grid size. Hence, further simula-tions were performed using 0.85 million cells and the kLawas found to be 0.042 s

1.

Figure 3 shows contours of kLa along a diametricplane x 0.0625 along the centerline. As is to beexpected, the highest value of kLa is found near theimpeller where turbulent dissipation rate is the highest.

The value of kLa for a single bubble size (0.042 s1) is

found to be significantly different from the experimental

results (0.0169 s1).32 This is because, as per discussion

above, the single bubble model does not account forbreakage and coalescence of bubbles. Gas volume frac-

tion contours shown in Figure 4 suggest that the rotating

impeller is not able to disperse the gas toward the tank

walls and buoyancy forces the gas toward the tank top

along the central shaft.

Population balance and model validation

To improve upon the accuracy of the simulation, popula-

tion balance equations (Eqs. 1019) are applied and method

of classes is used. Different number of bins were tried in the

model and based on the results, 13 bins were selected with

the minimum bin size equal to 0.00075 m. Smaller number

Figure 2. Plot ofkLa as a function of mesh size.

The plot shows that grid independency is achieved at 0.85 mil-lion cells.

Figure 3. Contours of mass transfer coefficient kLa along x 50.0625 plane for model without using population bal-ance equations (PBE).

Figure 4. Contours of gas volume fraction along x 5 0.0625plane for model without using population balanceequations (PBE).

Table 1. Bubble Sizes for all 13 Bins Obtained by Using Methodof Classes (Discrete Model) to Solve Population BalanceEquations (PBE)

Bin Number Size (m)

Bin-0 0.012Bin-1 0.00952Bin-2 0.00755Bin-3 0.006Bin-4 0.00476Bin-5 0.00377Bin-6 0.003Bin-7 0.00238Bin-8 0.00188Bin-9 0.0015Bin-10 0.00119Bin-11 0.000944

Bin-12 0.00075



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resulted in significant errors in solution while larger number

offered a marginal increase in accuracy but a significant

increase in the time needed to arrive at the final solution.

Table 1 shows the bubble size (size boundary) for each of

the 13 bins. Volume fraction of Bin-6 (3 mm) is taken as

unity while defining the velocity inlet boundary condition for

discrete (gas) phase. Figure 5A shows kLa contours along x 0.0625 plane for the 13 bubble classes. The kLa value iscalculated to be 0.0174 s

1, which is quite closer to reported

experimental value of 0.0169 s1.32 Figure 5B shows contours

of Sauter-mean diameter d32 along the above mentioned

plane. The volume-averaged d32 value for the 13 classes is0.00361 m. It shows that coalescence dominates over break-

age and that the smallest bubbles are found at the bottom of

tank. Figure 5C shows the velocity profile of gas phase in the

reactor and it is seen that the velocity of gas is maximum

near the impeller and decreases as we move away from it.

Design of experiments and data analysis

Process characterization studies are performed to evaluate

process robustness and thereby determine design space

within which we can operate and obtain acceptable process

quality and consistency.19,37 DOE is used for evaluating the

effect of the various independent variables on kLa. In this ar-

ticle, full factorial design is used with three levels for gas

flow rate and impeller RPM and two levels for liquid level.

Table 2 (first three columns) shows the DOE that was per-

formed. Simulations are performed for each set of conditions

and kLa is calculated.

For data analysis, linear regression is used to relate kLawith the independent variables. The relationship is also

known as a regression equation and the general linear model

has the form:

y Xm

ibixi E (27)

where y is the dependent variable, bi is a constant, xi is anindependent variable, m is the total number of observations,and E represents random error. In this article, least squaresmethod is used for estimating bi and E by minimizing thesum of the squared difference between the actual y valuesand the predicted y values.

Table 2 also presents the kLa values for each set of chosenconditions. The results are analyzed via linear regression

using commercial software JMPVR . Summary of the model fit

is shown in Table 3. Here, R2 is the square of the correlationbetween the actual and predicted response. It estimates the

proportion of variation in the response around the mean that

can be attributed to the terms in the model rather than to

random error. Further, R2Adj is the ratio of mean squares and

it adjusts the R2, making it comparable across models with

Figure 5. Contours for model using 13 classes of bubble sizes.

(A) Mass transfer coefficient kLa along x 0.0625 plane; (B) Sauter-mean diameter d32 along x 0.0625 plane; (C) Gas velocity v along x 0.0625 plane.



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different numbers of parameters by using degrees of freedom

in its computation. Root mean square error is the square root

of the mean square for error in the corresponding analysis of

variance and estimates the standard deviation of the random

error. Mean of response is the overall mean of the responsevalues. Observations (or sum of Weights) record the number

of observations in the fit. If there are no missing and

excluded rows it is same as number of rows in data table. If

there is column associated with role of weight than it is sum

of the weight of column values. In our study, R2 for the fit isfound to be 0.97 and the R2Adj is found out to be 0.96. Thisshows that the model fits the data accurately.

Figure 6 presents the results from analysis of the DOE data.

Figures 6A,B show that all three parameters that are evaluated,

namely flow rate, agitation rate, and liquid level, have a statis-

tically significant impact on kLa with the flow rate having themost impact and the liquid level the least. Further, as seen in

Figures 6A,C, second order interactions between the three pa-rameters are also statistically significant, although the interac-

tions are smaller in magnitude than the main effects. The final

equation for the overall model is as follows:

kLa 0:04095 0:0073667 Flow Rate 2:85e5

RPM 0:17361 Liquid Level Flow Rate 1:5

RPM 600 8:125e6 Flow Rate 1:5

Liquid Level 0:16 0:11833 RPM 600

Liquid Level 0:16 3:625e4 28

Equation 28 is an empirical expression that was obtained

from statistical analysis of the data obtained from DOE

results. Since it is not a mechanistic model, the expression

will only be valid for the system in question and in the

ranges of the independent parameters that are examined in

this study. The equation should not be used to extrapolate

outside the characterization space that is examined.

Estimation of design space for mixing

Design space has been defined as The multidimensional

combination and interaction of input variables (e.g., material

attributes) and process parameters that have been demonstrated

to provide assurance of quality. Working within the design

space is not considered as a change. Movement out of thedesign space is considered to be a change and would normally

initiate a regulatory post-approval change process. Design space

is proposed by the applicant and is subject to regulatory assess-

ment and approval.13 Once the process has been modelled, as

in Eq. 28, design space can be conveniently established.

In this article, we are establishing the design space for

mixing, quantified as kLa of 0.0150.02 s1. Traditionally,

we are used to associating design space with a rectangular

box within which all combinations of conditions will yield

the desired output (in this case kLa). However, this oversim-plified visualization only applies for the case when we have

a two variable system and there are no interactions between

them. The system that we are examining in this article is

complex on both fronts. Not only we have three different

Table 2. CFD Simulations Performed as Per a Full-Factorial DOEwith Gas Flow Rate and Agitation Rate (RPM) at Three Levels andLiquid Level at Two Levels*

Gas flowrate 105

(m3

s1

)

AgitationRate

(RPM)Liquid

Level (m)kLa(s

1)

1.5 600 0.18 0.01740.5 600 0.18 0.00922.5 600 0.18 0.0321

1.5 400 0.18 0.01241.5 800 0.18 0.02450.5 400 0.18 0.00310.5 800 0.18 0.01372.5 400 0.18 0.01622.5 800 0.18 0.03611.5 600 0.14 0.01130.5 600 0.14 0.00652.5 600 0.14 0.01681.5 400 0.14 0.00861.5 800 0.14 0.01530.5 400 0.14 0.00230.5 800 0.14 0.00972.5 400 0.14 0.01032.5 800 0.14 0.0214

*The Calculated kLa Values are Also Listed for Each Simulation.

Table 3. Summary of Fit for the Empirical Model Created fromStatistical Analysis of the DOE Results

R2 0.97428R2Adj 0.960251Root Mean Square error 0.00181Mean of Response 0.014828Observations (or Sum of Weights) 18 Figure 6. Summary of statistical analysis of DOE data.

(A) Sorted parameter estimates showing the statistical signifi-cance and magnitude of the effect of the various parameters onkLa. It is seen that flow rate is the most significant parameter.(B) Prediction profiler illustrating the main effects of the threeparameters examined. Results corroborate observations fromFigure 6A. (C) Interaction profiles showing the interactionsamongst the three parameters examined. It is seen that all three

parameters interact with each other.



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variables that are significantly impacting kLa but also theirinteractions are significant.

The easiest way to envision design space in a case such as

ours is by deciding on an acceptable range for one of the

three parameters and then using the model to establish design

space for the remaining two parameters. In our case, the

design space for liquid level (the least significant parameter)

is assumed to be 0.160.17 m. Figure 7A illustrates the varia-

tion in kLa at liquid level of 0.16 m when varying the remain-ing two parameters, agitation rate, and flow rate. It is seen

that infinite number of ranges can be chosen for each of thetwo parameters (infinite rectangles can be drawn within the

white space in Figure 7A). One such design space is given in

Table 4 (flow rate 1.51.9 105 m3 s1 and agitation rate600660 rpm). This is also shown as the filled box in Figure

7A. It must be emphasized again that any number of such

combinations can be chosen to meet the overall requirement

of kLa. Also, it is evident that greater flexibility in one param-eter will result in narrower range for the other parameters.

For example, for the illustration in Figure 7A, if greater flexi-

bility in flow rate is desired (say 1.52.1 105 m3 s1)then the agitation rate can only vary between 600620 rpm

(shown as an empty box in Figure 7A).

Figure 7B illustrates the same information as in Figure 7A

but at the liquid level of 0.17 m. Once again, many combi-

nations of flow rate and agitation rate can yield a kLabetween 0.015 and 0.02 s

1. One such combination is listed

in Table 4 (flow rate 1.31.6 105 m3 s1 and agitationrate 600670 rpm). Based on the chosen set of ranges in Ta-

ble 4, an overall design space can be established for liquid

flow (0.160.17 m), flow rate (1.51.6 105 m3 s1), and

agitation rate (600660 rpm). Operating anywhere withinthis cube will result in a kLa of 0.0150.02 s

1. It can be

seen that the chosen design space is a bit more restrictive on

flow rate and liquid flow and more liberal on the agitation

rate. As mentioned earlier, a different combination can be

chosen if greater flexibility is desired in one parameter vs.

others.

Conclusions

We have demonstrated the usefulness of CFD both for

modeling of complex applications such as mixing in a bio-

reactor and for evaluating the effect of the different input pa-

rameters on the mixing. In combination with statistical

concepts of design of experiments, design space can beestablished for the input parameters. Chosen experiments can

then be performed (at edges of the model) to validate the

results and if successfully validated, considerable savings

can be realized in comparison with the traditional approach

where 18 experiments (as per Table 2) would be otherwise

needed to generate the same process knowledge. We hope

that the article encourages other scientists in academia and

industry to adopt CFD for solving complex problems that

are so often faced in biotechnology process development.

Acknowledgments

The authors thank Mr. Shitalkumar Joshi and Mr. Sravanku-

mar Nallamothu from Ansys-Fluent India for technical help

and discussions.

Notation

Ui mean velocity of i (liquid or gas) phase (m s1)

p pressure (N m2)Ri momentum exchange in rotating frame (N m

3)Fi centrifugal forces in rotating frame (N m

3)g acceleration due to gravity (9.81 m s2)

CD drag coefficientReP Reynolds number

d bubble diameter (m)C model parameter accounting for effect of turbu-

lence in slip velocity; assigned value of 0.3

k turbulent kinetic energy (m2 s3)

Figure 7. Contour profilers showing the design space for flowrate and agitation rate for achieving kLa of 0.0150.02 s21.

The filled rectangles show the defined design space. (A) Designspace at liquid level of 0.16 m. (B) Design space at liquid levelof 0.17 m.

Table 4. Example of a Design Space for Achieving kLa of0.0150.02 s21

AgitationRate (RPM)

FlowRate 105

(m3

s1

)

Design space corresponding toliquid level of 0.16 m

600660 1.51.9

Design space corresponding toliquid level of 0.17 m

600670 1.31.6

Final design space Liquid level0.160.17 m,

Flow rate1.51.6 105 m3 s1,

and agitation rate600660 rpm.



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GkL rate of production of turbulent kinetic energy(m2 s4)

Cl, C1e,C2e, rk, re

model parameters having values equal to 0.09,1.44, 1.92, 1.0, and 1.3, respectively

ni number of bubbles in the ith bubble class

di diameter of bubbles in the ith bubble class (m)

BiB birth rate due to breakage (m3 s1)

BiC birth rate due to coalescence (m3 s1)

DiB death rate due to breakage (m3 s1)

DiC death rate due to coalescence (m

3

s

1

)a(V,V0) coalescence rate between bubbles of size V and V0

(s1)b(V0) breakage rate of bubble with size V (s1)

m(V0) number of daughter bubbles formed due to break-age of size V0

n(V, t) number of bubbles of size V at time tp(V,V0) probability density function for bubbles of size V,

generated from bubble of size V0

fi ratio of volume fraction of ith group bubbles and

total gas volume fractiond32 Sauter mean diameter (m)Cf increase in surface area due to breakage

Pc coalescence frequencydB bubble diameter used while performing simulation

without PBE (3 mm)

dH sparger hole diameter (m)kL local liquid mass transfer coefficient (m s1)

a interfacial area available for mass transfer (m2)DO2 diffusion coefficient of oxygen (1.97 10

9

m2 s1)y dependent variable in linear regression equation

xi independent variable in linear regression equationm total number of observations taken for linear

regressionE random error in linear regression equation

Greek letters

ai volume fraction of i (liquid or gas) phasebi constant used in linear regression equatione turbulent dissipation energy (m2 s3)

yij collision rate function (m3 s1)

k arriving eddy size (m)lG viscosity of air (1.7894 105 kg m1 s1)

lL viscosity of water (0.001 kg m1 s1)

lt,L liquid phase viscosity (kg m1 s1)

k dimensionless eddy sizePkL and

PeL

terms accounting for influence of continuous phaseon dispersed phase

qi density of i (liquid or gas) phase (kg m3)

rL surface tension of water (0.0719 N m1)

sef stress tensor (kg m1 s2)

XB(Vj, Vi) breakage rate of bubble size Vj in bubbles of sizeVi and Vj Vi (m

3 s1)

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Manuscript received July 11, 2011, and revision received Oct. 14, 2011.


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