Chemical Engineering Science 95 (2013) 257–266

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Chemical Engineering Science

0009-25http://d

n CorrE-m

my (J. N

journal homepage: www.elsevier.com/locate/ces

Multi-scale models for the optimization of batch bioreactors

Emily Wan-Teng Liew, Jobrun Nandong, Yudi Samyudia n

Department of Chemical Engineering, Curtin University, CDT 250, 98009 Miri, Sarawak, Malaysia

H I G H L I G H T S

� A new approach to the bioreactor modelling is proposed.

� Mixing effects are included in the modelling and optimization of bioreactors.� The model is used to predict the yield and productivity of the batch fermentation process.

a r t i c l e i n f o

Article history:Received 18 November 2011Received in revised form22 February 2013Accepted 15 March 2013Available online 29 March 2013

Keywords:BioreactorCFDFermentationKineticsModellingOptimisation

09/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.ces.2013.03.036

esponding author. Tel.: +60 85 443939; fax: +ail addresses: emily.liew@curtin.edu.my (E.-T.andong), yudi.samyudia@curtin.edu.my, samy

a b s t r a c t

Process models play an important role in the bioreactor design, optimisation and control. In previouswork, the bioreactor models have mainly been developed by considering the microbial kinetics and thereactor environmental conditions with the assumption that the ideal mixing occurs inside the reactor.This assumption is relatively difficult to meet in the practical applications. In this paper, we propose anew approach to the bioreactor modelling by expanding the so-called Herbert's microbial kinetics (HMK)model so that the developed models are able to incorporate the mixing effects via the inclusion of theaeration rate and stirrer speed into the microbial kinetics. The expanded models of Herbert's microbialkinetics allow us to optimize the bioreactor's performances with respects to the aeration rate and stirrerspeed as the decision variables, where this optimisation is not possible using the original HMK model ofmicrobial kinetics. Simulation and experimental studies on a batch ethanolic fermentation demonstratesthe use of the expanded HMK models for the optimisation of bioreactor's performances. It is shown thatthe integration of the expanded HMK model with the computational fluid dynamics (CFD) model ofmixing, which we call it as a kinetics multi-scale (KMS) model, is able to predict the experimental valuesof yield and productivity of the batch fermentation process accurately (with less than 5% errors).

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Bioreactors are widely used in process industries for mixingand blending of liquids for biochemical reactions (Harvey andRogers, 1996; Rahimi and Parvareh, 2005). Bioreactors used inbiotechnological applications are commonly designed to meet therequirements of the microorganism cell-culture environment byaddressing the key variables such as temperature, oxygen, pH,nutrients, metabolites and biologically active molecules(Hutmacher and Singh, 2008). The bioreactor's performances arecharacterized by its transport capacities in order to optimallysupply the microorganism with the required nutrients during thefermentation process (Lübbert, 1992).

It is very common that the bioreactors are equipped withimpellers such that they operate in the turbulent flow regime in

ll rights reserved.

60 85 443837.Liew), jobrun.n@curtin.edu.udia@yahoo.ca (Y. Samyudia).

order to improve the mixing conditions. The presence of such mixingconditions often makes the task of bioreactor optimisation difficult(Ranade, 1997). It has been recognized that the awareness of the non-uniformity distribution of the intensity and quality of flow, turbulentkinetic energy, turbulent eddies and concentrations of speciesinvolved throughout the bioreactor is essential to devise an efficientoperational strategies of bioreactors, not only to achieve good yieldbut also consistent product quality (Venneker et al., 2002).

From systems engineering perspective, mathematical model-ling has been one of the most successful scientific tools available toimprove the performance of a bioreactor via the improvement ofthe metabolic capabilities of microorganism by mean of geneticmanipulations of the cell metabolism and the bioprocess condi-tions (Wiechert, 2002). Hence, process models would becomemore pervasive in the design, optimisation and control of bior-eactors (Jiang et al., 2002).

The common approach in the bioreactor design, optimisationand control has always relied on the kinetics of fermentation,which assumes well-mixing behaviour. Our previous studies, forexample Liew et al. (2009) showed that the deviation from the

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266258

ideal mixing behaviour (i.e. non-ideal mixing phenomena) couldlead to severe loss in yield and changes in microbial physiology.The integration of mixing phenomena into the bioreactor model-ling is therefore vital, but it is not an easy task because the detaileddescription of the turbulent flow-field, in combination with othertransport equations, needs to be addressed for the interactions ofmixing and fluid flow (Jenne and Reuss, 1999).

The mixing is particularly important in the case of suchpractical applications as the industrial fermentation processes(Bezzo et al., 2003). It is also important to note that such afermentation process can be highly sensitive to other variablessuch as batch time, liquid volume and initial nutrient concentra-tions due to their effects on cellular metabolism. The optimisationof bioreactors now implies the manipulation of both microbialculture and the environmental factors involved, i.e. a multivariableoptimisation of the process (Konde and Modak, 2007). Model-based optimisation is therefore a vital tool in determining thebatch operating strategies (Hjersted and Henson, 2006). One of thefundamental aspects to the success of the model-based optimisa-tion is the model of microbial kinetics adopted, which mustaccurately capture the effects of decision variables (pH, tempera-ture, aeration rate and stirrer speed) on the rates of growth,substrate consumption and product formation.

To date, most of the currently available models of microbialkinetics do not capture the effects of the aeration rate (AR) andstirrer speed (SS) on the rates of growth, substrate consumptionand product formation. Therefore, the main contribution of thispaper lies in the development of bioreactor models, where threeapproaches to incorporating the effects of AR and SS into themodel of microbial kinetics based on the Herbert's concepts areproposed. Two of the three developed models are referred to inthis paper as the expanded Herbert's microbial kinetics (HMK)models. The proposed approaches can also be extended to anyother conventional model of microbial kinetics if we want toincorporate the fluid mixing phenomena.

The objectives of this paper are two folds. The first objective is topropose three modelling approaches to capturing the mixingmechanism so that AR and SS can be used as parameters to optimizethe bioreactor's performances, i.e. yield and productivity. The secondobjective is to determine the optimal values of AR and SS for a batchfermentation process by applying the response surface methodology(RSM) to maximize the fermentation yield and productivity using thedeveloped models. Simulation and experimental studies were per-formed to validate the results of this work.

Table 1Input variables and their levels employed in two-factor factorial design.

Factor Variable Units Low level(−)

Middle level(0)

High level(+)

X1 Aeration rate(AR)

LPM 1.0 1.25 1.5

X2 Stirrer speed(SS)

rpm 150 200 250

2. Material and methods

2.1. Microorganism and inoculum

In this study, a set of experiments was conducted using theBIOSTAT A-plus 2L, MO-Assembly bioreactor operated at batch mode.Rushton turbine is used as the agitation system, whereas air spargeris utilized as the aeration system. Rushton turbine is a disc turbine,which has been considered as the optimum design for use in manyfermentation processes (Stanbury et al., 1995). This kind of discturbine is most suitable in a bioreactor since it could break up a fastair stream without itself becoming flooded in air bubbles. Theindustrial Baker's yeast was utilized as the inoculum culture withglucose as the substrate. The inoculum was grown in a 250 mLconical flask and was incubated at room temperature for 8 h.

2.2. Growth medium

First, 1.5 L of fermentation medium was prepared by adding75 g glucose, 7.5 g yeast, 3.75 g NH4Cl, 4.37 g Na2HPO4, 4.5 g

KH2PO4, 0.38 g MgSO4, 0.12 g CaCl2, 6.45 g citric acid and 4.5 gsodium citrate. The medium culture was sterilized at 121 1C for20 min and then cooled down to room temperature. Then, 40 mLof inoculum was added to the fermentation medium. The tem-perature and pH were maintained at 30 1C and pH 5, respectively.The batch process was stopped after 72 h and the samples weretaken at every 2–4 h of sampling interval and were analyzed forglucose and ethanol concentrations. The experiments wererepeated at various values of AR and SS within the range of1.0–1.5 LPM of AR and 150–250 rpm of SS i.e. ½1:0;150:0�T≤AR SS

� �T≤½1:5;250:0�T . Therefore, the base-line condition orthe mid-point of these ranges of AR and SS is at 1.25 LPM of ARand 200 rpm of SS.

2.3. Glucose and ethanol concentrations

Each sample was first filtered, and then analyzed for theconcentrations of substrate and ethanol using R-Biopharm testkits and UV spectrophotometer under a wavelength of 340 nm, asoutlined in the procedures provided by the test kits. All sampleswere tested at room temperature.

2.4. Optical density

Each sample was first filtered. The optical density was thenanalyzed by using UV spectrophotometer under a wavelength of340 nm, similar wavelength utilized for the analysis of glucose andethanol concentrations, in order to show consistency of concen-trations analysis. No test kits were required as the UV spectro-photometer could directly analyze the optical density. All sampleswere tested under room temperature.

2.5. Experiment designs for response surface methodology

In this study, we adopt the central composite design (CCD) forthe design experiments since this design provides a solid founda-tion for the generation of a response surface map. To create a CCD,it is important to locate new points along the axes of the factorspace. Table 1 shows the CCD matrix employed for both AR and SSand the corresponding values of yield and productivity. Formaximum efficiency, the axial or star points are to be located ata specific distance outside the original factor range. In theapplication of Response Surface Methodology, the regressionanalysis is employed to describe the experimental data collected.The least square technique is used to fit the model equationcontaining the input variables by minimizing the residual errorsof the sum of squared deviations between the experiments andthe estimated responses.

In our study, a two-factor factorial design is selected since ARand SS are the two factors of interest to study on the effect ofbioreactor performance. AR (X1, LPM) and SS (X2, rpm) areconsidered as input variables. Yield (Y1, %) and productivity (Y2,g/L h) are considered as output variables. The levels of the inputvariables are selected based on the range of reasonable formula-tions since the interpretation of the results are valid only within

Table 2CCD matrix for the two independent variables (aeration rate and stirrer speed).

Standardorder

Runorder

Block X1: aerationrate (LPM)

X2: stirrerspeed (rpm)

Y1:yield(%)

Y2:productivity(g/L h)

7 1 1 1.25 200 21.500 0.1801 2 1 1.0 150 14.788 0.0995 3 1 1.25 200 21.050 0.1766 4 1 1.25 200 21.250 0.1783 5 1 1.0 250 15.105 0.1024 6 1 1.5 250 24.040 0.1602 7 1 1.5 150 16.392 0.10613 8 2 1.25 200 24.000 0.23012 9 2 1.25 200 23.500 0.20014 10 2 1.25 200 22.000 0.19010 11 2 1.25 129.29 18.511 0.1159 12 2 1.60 200 22.250 0.19511 13 2 1.25 270.71 23.500 0.2108 14 2 0.90 200 20.500 0.165

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266 259

experimental limits in the laboratory available. Three levels arecoded as −1, 0 and +1, which corresponded to the lower, middleand higher values respectively. The experiments are chosen torealize every possible combination between the variables, with thelevels coded. Therefore, Table 1 shows the input variables andlevels employed.

The experiments are randomized in order to make the experi-mental error as small as possible. For CCD, it is an extension oftwo-level full factorial design. A CCD enables a quadratic model tobe fitted by including new levels. Therefore, there are 2 blocksavailable in the CCD matrix shown in Table 2. Once all experimentsin Block 1 has been completed and all data were recorded, it isrequired to extend the experiments to Block 2. All experimentaldata from Block 1 and Block 2 will then be used to develop themathematical model.

As shown in Table 2, the experimental tests involved 14 trials.Standard Order is the order of treatment combinations based onthe level indicated in Table 2. Run Order is the order of experi-ments to be carried out. For each experimental trial, the newconditions of the AR and SS were utilized. These results werefurther analyzed by performing the ANOVA on the residuals fordetecting outliers (Noordin et al., 2004).

3. Bioreactor modelling

3.1. Objectives and approaches

The majority kinetics of ethanol fermentation utilize a formalmacro-scale approach to describe the microbial growth, wherebythey are empirical and based on either Monod's equation or on itsnumerous modifications which take into account the inhibition ofmicrobial growth by a high concentration of product and/orsubstrate (Starzak et al., 1994). The models so far only explainedthe effect of ethanol inhibition via the mechanism of non-competitive inhibition of a simple reversible enzymatic reactionwithout taking into consideration of the mixing mechanismoccurring inside the bioreactor, i.e. the assumption of well-mixing behaviour is applied. Deviation from the ideal mixingbehaviour in practice could lead to severe loss in yield and changesin microbial physiology. Thus, the integration of mixing phenom-ena (and the effects of AR and SS) is necessary to be taken intoaccount, and this is the objective of our modelling work.

Three modelling approaches to incorporating the effects ofmixing in a batch ethanolic fermentation process are thereforeproposed in this paper: (1) statistical data-based (SDB) model,(2) kinetics hybrid (KH) model and (3) kinetics multi-scale (KMS)

model. To validate the developed models, a series of experimentalstudies were conducted using the BIOSTAT A-plus 2L, MO-Assembly bioreactor operated at batch mode with the ranges ofAR of 1.0–1.5 LPM and SS of 150–250 rpm.

3.2. Batch bioreactor model with kinetics Herbert's concept

In order to predict the yield Y and productivity Pr , a batchbioreactor can be modelled dynamically as follows:

_Z ¼_Xv_S_P

264

375¼

rxrsrp

264

375 ð1Þ

Φ¼Y

Pr

" #¼

100%ðPðtbÞ−P0Þ=ðS0−SðtbÞÞðPðtbÞ−P0Þ=tb

" #ð2Þ

where _Z is the vector of state variables consisting of optical density( _Xv), substrate (_S) and product ( _P) concentrations. Φ is the vectorof performance measure consisting of Y and Pr .

S0 ¼ Sð0Þ and P0 ¼ Pð0Þ is the initial substrate and ethanolconcentrations (g/L) of the medium, (tb) (h) is the batch time forthe fermentation process. Other variables are the concentrationprofiles of substrate (S), product (ethanol) (P) and viable cell(optical density) (Xv). The microbial kinetics are: (1) rate of growth(rx), (2) rate of product formation (rp) and (3) rate of substrateconsumption (rs).

In this work, the Herbert's concept of endogenous metabolismis adopted since it has been used in numerous studies to describethe kinetics of ethanolic fermentation with sufficient accuracy(Starzak et al., 1994). Moreover, the by-product concentration isnot included in this study because we focus on the impacts of ARand SS on the concentration profiles of substrate S, product(ethanol) P and viable cell (optical density) Xv.

The Herbert's concept assumes that the observed rate ofbiomass formation (rx) comprised of the growth rate ðrxÞg andthe rate of endogenous metabolism ðrxÞend.rx ¼ ðrxÞg þ ðrxÞend ð3Þ

where

ðrxÞg ¼ ½k1XvS=ðk2 þ SÞ�expð−k5PÞ ð4Þ

ðrxÞend ¼−k6Xv ð5ÞThe rates of substrate consumption (rs) and product formation

(rp) are assumed to be proportional to the biomass growth rate:

rs ¼−k3ðrxÞg ð6Þ

rp ¼ k4ðrxÞg ð7Þ

Note that the kinetics of ethanolic fermentation based on theHerbert's concept consists of six kinetic model parameters i.e.Κ ¼ k1 k2 k3 k4 k5 k6�T

h.

3.3. Statistical data-based (SDB) model

Rather than using the bioreactor models equations from (1–7),the SDB model is developed by applying a regression analysis to aset of experimental data for different AR, SS, yields (Y) andproductivity (Pr). By applying this approach, the effect of mixingarising from different values of AR and SS are included in theregression model by treating the AR and SS as inputs (or experi-mental variables X) and the yield ( QUOTE ) and productivity (Pr)as outputs (or response variables Φ. After applying design experi-ments to the bioreactor, sets of experiment data for both inputsand outputs are obtained.

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266260

Generally in the development of statistical-based model, it isassumed that there exists a relationship ℱ : X-ℱ where X∈Rn isan n-dimensional vector of inputs, Φ∈Rm is an m-dimensionalvector of outputs and ℱ∈Rm is an m-dimensional vector offunctions space. As Φ is normally an implicit function of X, in realpractice it is often difficult to obtain the exact relationshipbetween the input and response vectors especially for complexsystems. Thus, one way to develop an explicit relationshipbetween them is by using a regression model where the modelparameters are obtained by an optimisation.

Let us consider the regression model as a quadratic model:

Φ¼ Aþ BX þ XTDX þ ε ð8Þwhere A, B, D are defined as model parameters, whereby A, B and Dwill be estimated in such a way that the sum of the squared errorsbetween the predicted performance measure (Φ) and experimen-tal values (Φ) of the responses are minimised. X is the vector ofexperimental inputs (AR and SS). ε is the predicted error. So, thisproblem can be mathematically stated as

P1 : min∑ki ¼ 1½ðΦ−ΦiÞ�T ½ðΦi−ΦiÞ� ð9Þ

where Φi is the predicted values by Eq. (8) and the subscript iindicates the experimental number. Based on the full factorialdesign, the total number of experimental runs for n inputs is givenby k¼ 2n; in our case n¼ 2 and hence k¼ 4. Note that differentvalues of k result in different experimental designs.

To test the optimisation results of P1, the analysis of variance(ANOVA) is performed where the results must be significant alongwith the analysis of curvature. Curvature analysis is vital to indicatewhether the experimental results could fit well into the proposedmodel (Wadsworth, 1998). If the curvature is significant, i.e. thecurvature lies in the region of the desired optimum response, theoptimisation results are acceptable. If the curvature is insignificant, theoptimisation results are not acceptable. A method based on the path ofsteepest ascent (PSA) (Wadsworth, 1998) is adopted, and where thecurvature is further analyzed until it is shown to be significant.

Note that some process constraints such as Xmin≤X≤Xmax

where Xmin and Xmax denote the lower and upper limits of inputs,respectively can be included in the optimisation of P1. Further-more, different model structures can also be selected while solvingthe optimisation such as linear or second order model as given by

Φ¼ Aþ BX þ ε; or ð10Þ

Φ¼ Aþ BX þ XXTCTε ð11Þ

where Φ is the predicted values and C is an off-diagonal matrix

defined as C : ¼0 c1c2 0

" #.

3.4. Kinetics hybrid (KH) model

The basic assumption underlying the development of kineticshybrid model (Starzak et al., 1994) is that the kinetic parametersare the function of the inputs X as

Κ ¼ hðX;ΘÞ ð12Þhere Θ∈Rnθ�mθ is a matrix whose elements correspond to theparameters to be determined later. Therefore substituting Eq.(12) into Eqs. (3)–(7), the microbial kinetics of Herbert's can nowbe expressed as

Rx ¼ gmðZ;X;ΚÞ ð13Þwhere Rx is the vector of microbial kinetics, i.e. rx, (rx)g and (rx)end.gm is the Herbert Microbial Model and X is the process constraint.

The advantage of expanded kinetics Herbert's model, i.e. Eq.(13) over the original microbial kinetic model, i.e. Eqs. (3)–(7) is

that the expanded one can be directly used to optimize the yield Yand the productivity Pr with respect to the aeration rate (AR) andstirrer speed (SS).

The development of kinetics hybrid model involves two mainsteps:

1.

For experimental run i, obtain the kinetics parameters κi usingthe original kinetics Herbert's model based on Eqs. (3)–(7) andbatch reactor model based on Eqs. (1) and (2)

2.

For the obtained kinetic parameters Κ i; ∀i∈1;2;⋯k and sets ofaeration rates (ARi) and stirrer speeds (SSi), find Θ in Eq. (12)using the regression method

The combination of the batch bioreactor model which isdenoted by Eqs. (1) and (2), the Herbert's kinetics model, denotedby Eqs. (3)–(7) and the regression model of Eq. (12) constitutes theso-called kinetics hybrid (KH) model of bioreactor. Clearly, in thisapproach, the effect of mixing is now embedded into the bior-eactor model, i.e. Eq. (1).

In more details, the development of kinetics hybrid modelfollows the systematic procedure as follows:

Step 1: Identification of Herbert's kinetic parameters.The Herbert's kinetic parameters (k1; k2;⋯; k6) are obtained viaoptimisation by solving the following quadratic problem:

P2 : minΚΣqj ¼ 1½Zj−Zj� ð14Þ

here Zj is the predicted value of Z using the bioreactor model ofEqs. (1) and (2) at the j-sample time, and q is the number ofsamples taken during the course of batch experiments.For the i-experimental run, the corresponding solution toproblem P2 will yield κi ¼ κni that minimizes the sum of squarederrors between the predicted values and experimental values ofZi. For a kþ 1 number of experimental runs, we will obtain aset of Κ ¼ kn0; k

n

1; kn

2;⋯; knk , which contains the solutions corre-sponding to all experimental runs including the base-lineexperimental run i.e. kn0.Step 2: Determine the regression model of Eq. (11).This step is to find the regression model of κ in term of X. Thus,this is equivalent to finding Θ¼Θn where the problem can bestated as

P2 : minΘΣji ¼ 0½κn1−κi�T ½κn2−κi�; ∀κi∈Κ ð15Þ

where j is the number of experiment runs based on the designof experiment (i.e. j¼4 if factorial design is adopted). Here i¼0,indicates the experiment at the base-line conditions and κdenotes the predicted value of κ based on a regression model, e.g. as Eqs. (8), (10) and (11). As we have no a priori knowledge onthe exact form of relationship h : X-κ, we use the statisticalapproach, i.e. the technique used for the SDB model, thusassuming κ can be represented by model equations e.g. byEq. (10) for linear model.

3.5. Kinetics multi-scale (KMS) model

The kinetics multi-scale (KMS) model is developed based onthe kinetics hybrid model described by Eqs. (3)–(7) and Eqs. (12)and (13), but we use the general mass–energy balance over anelement of reactor volume combined with a mixing model toreplace the bioreactor model which is denoted by Eqs. (1) and (2).The mixing model is implemented using computational fluiddynamics (CFD) based on the k−ε turbulence model (Dubeyet al., 2006). This approach was used to describe the mixingmechanism in a bioreactor with sufficient accuracy (Ranade,2002).

Table 3ANOVA results on yield for SDB model.

Source p-Value prob4F

Model 0.0047 SignificantA−AR 0.0117B−SS 0.0059AB 0.0309A2 0.0341B2 0.0201ResidualLack of fit 0.0563 Not significant

Table 4ANOVA results on productivity for SDB model.

Source p-Value prob4F

Model 0.0054 SignificantA−AR 0.0718B−SS 0.0069AB 0.1978A2 0.0203B2 0.0035ResidualLack of fit 0.2461 Not significant

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266 261

The k−ε turbulence model is normally used in order to describethe mixing behaviour and to compute the turbulence in thebioreactor. The energy dissipation is expressed as

ε¼ ððΔPFuÞÞ=m¼ ðΔPuÞ=ðxρÞ ð16Þwhere Δp denotes the pressure drop, m the mass, F the tube cross-section, x the axial coordinate and u the velocity vector field.Moreover, the fluid flow equations need to be solved for a constantdensity fluid (Bode, 1994). These consist of the continuity equa-tion:

divðρuÞ ¼ 0 ð17Þand the transport equations:

divðρukÞ ¼ div½ðμeff=skÞ∇k� þ G−ρε ð18Þ

divðρukÞ ¼ div½ðμeff=sεÞ∇ε� þ ðC1G−C2ρεÞðε=kÞ ð19Þwhere ∇k and ∇ε denote the gradient of k and ε, respectively. k isthe kinetic energy of turbulence at the point of interest.

On the other hand, the Eddy viscosity model is used to solvecomplex turbulent flows. This model has proven to be a valuabletool in the predictions of turbulent flow-field (Gatski and Jongen,2000). Therefore, it would be useful to utilize this model to predictthe turbulent flow-field in the bioreactor.

The Eddy viscosity is given by

uT ¼ Cμρk2=ε ð20Þ

Note that G is the scalar dissipation function G¼ τijτij=ð2μeff Þand the scalar values: Cμ ¼ 0:09; C1 ¼ 1:44; C2 ¼ 1:92; sk ¼ 1 andsε ¼ 1:3.

The Navier–Stokes equation is used for flow equations todescribe the instantaneous behaviour of the turbulent liquid flowin ethanolic fermentation process. The resulting Reynolds equa-tions and the continuity equation are given by

∂ðρuiÞ∂t

þ ∂ðρuiujÞ∂xi

¼ −∂∂xi

ðτij þ ρu′iu

′jÞ−

∂p∂xi

þ ρgi ð21Þ

∂ρ∂t

þ ∂∂xi

ðρuiÞ ¼ 0 ð22Þ

For model accuracy and computational expense, a reasonableEddy viscosity models relating the individual Reynolds stresses tomean flow gradients are adopted:

ρu′iu

′j ¼−ρvturbð

∂ui

∂xjþ ∂uj

∂xiÞ þ 2

3ρ∂ijk ð23Þ

where vturb is the turbulent Eddy viscosity. The transport ofmomentum is thought of as turbulent eddies, which like mole-cules, collide and exchange momentum.

The general balance over an element of reactor volume is givenby

δðρϕÞ=δt þ δðρUiϕÞ=δxi ¼ δðΓϕðδϕÞ=δxiÞ=δxi þ Sϕ ð24Þwhere ρ is the density of fluid, ϕ is the concentration of anycomponent, Ui is the local velocity in the xi—direction, Γϕ is theeffective diffusivity of ϕ and Sϕ is the volumetric source term (rateof production of ϕ per unit volume). Note that, the reaction ratesdescribed by Eq. (13) are embedded into the source term Sϕ. Alsonote that the notation ϕ≡Z denotes the output variables i.e. opticaldensity, substrate and ethanol concentrations.

Using the KMS model, we can compute the mass of substrate(MStb ) and mass of product (MStb ) at the end of batch time overthe total reactor volume, i.e.

MStb ¼∑hi ¼ 1ðSiΔViÞ ð25Þ

MPtb ¼∑hi ¼ 1ðPiΔViÞ ð26Þ

where Si is the substrate concentration at i location, Vi is thevolume of mesh at i location and the volume of medium in thereactor VR is given by

VR ¼∑hi ¼ 1ΔVi ð27Þ

Here h is the number of discretizations (i.e. finite volume).Then, we can calculate as follows:

Φδ ¼Y

Pr

" #¼

100ðMPtb−MP0Þ=ðMS0−MStb ÞðMPtb−MP0Þ=tb

" #ð28Þ

here MS0 and MP0 correspond to the initial mass of substrate andmass of product (ethanol) in the fermentation medium,respectively.

3.6. Model analysis and validation

In this section, we develop the models using the proposedapproaches, then the models are analysed using the ANOVA andvalidated against the experimental data.

The SDB model was developed using the input and output dataas shown in Table 2, and the regression method results in thefollowing quadratic model (Eq. (29)) where the ANOVA analysis ispresented in Tables 3 and 4

Y

Pr

" #¼ 33:098

0:234

� �þ −18:785 −0:143

−0:139 −9:9� 10−4

� �ARSS

� �

þ AR SS� � 0 0:147

1:02� 10−3 0

� �ARSS

� �ð29Þ

As shown in Tables 3 and 4, the ANOVA results of the modeldemonstrate that the model is highly significant, as indicated bythe Fisher's F test (Fmodel¼9.73 and 9.3) and a low “Prob4F” value(Pmodel40.0047 and 40.0054). Additionally, the goodness of thefit of the model is also checked by the determination coefficient(R2). In this case, the values of the determination coefficient(R2¼0.8743 and R2¼0.8691) indicate that 87% of the samplevariation in yield and productivity are well explained by themodel. Thus, the SDB model is statistically adequate to predict

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

Batch Age (hr)

Act

ual E

than

ol C

once

ntra

tion

(g/L

sol

utio

n)

Exp. DataModel

Fig. 2. Model fitting for actual ethanol concentration (g/L solution) vs. batchage (h).

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266262

the yield and productivity within the range of experimentalsetting.

For the kinetics hybrid model, solving the optimisation pro-blem of P2 gives the predicted Herbert's kinetic parameters asshown in Table 5. This set of data is then used to obtain thefollowing regressed linear model by solving the optimisationproblem of P3

κ¼

k1k2k3k4k5k6

26666666664

37777777775¼

1:4085−0:2852X1þ 0:3692X20:00100:6631−0:0148X1þ 0:0220X20:1040þ 0:0142X1þ 0:0128X20:7558−0:1019X1−0:0211X20:0143−0:0001X1−0:0019X2

2666666664

3777777775

ð30Þ

By combining the linear model of Eq. (30) with the macro-scalebioreactor of Eqs. (1)–(7), the kinetics hybrid (KH) model can beobtained. This KH model is then validated against another set ofexperimental data of AR and SS, which were chosen within theexperimental ranges of AR and SS, i.e. 1.2 LPM AR and 175 rpm SS.Figs. 1–3 show the results of model validation–prediction usingthe KH model as compared to experimental data. As observed, thekinetics hybrid model can fit the experimental data well.

Similarly, a series of simulations were performed by using theRunge Kutta method to validate the KMS model, where a set of ARand SS was generated using the CCD technique. The ANOVA resultsof the fitness of the KMS model are shown in Tables 6 and 7.

The determination coefficients of the KMS model areR2¼0.8353 and R2¼0.8021, respectively for the yield and produc-tivity. This result shows that more than 80% of the samplevariation in yield and productivity are well explained by the

Table 5Predicted Herbert's kinetics parameters.

Runorder

X1: AR(LPM)

X2: SS(rpm)

k1 k2 k3 k4 k5 k6

1 1.25 200 1.4085 0.0010 0.6631 0.1040 0.7558 0.01432 1.0 150 1.3245 0.0010 0.6559 0.0770 0.8788 0.01633 1.25 200 1.1257 0.0010 0.6533 0.0909 0.7252 0.01734 1.25 200 1.2591 0.0010 0.6731 0.0879 0.7127 0.01795 1.0 250 2.0629 0.0010 0.6999 0.1026 0.8366 0.01256 1.5 250 1.4925 0.0010 0.6703 0.1310 0.6328 0.0123

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

50

Batch Age (hr)

Act

ual G

luco

se C

once

ntra

tion

(g/L

sol

utio

n) Exp. DataModel

Fig. 1. Model fitting for actual glucose concentration (g/L solution) vs. batchage (h).

Fig. 3. Model fitting for optical density vs. batch age (h).

Table 6ANOVA results on yield for the KMS model.

Source p-Value prob4F

Model 0.0115 SignificantResidualLack of fit 0.2003 Not significantPure error

Table 7ANOVA results on productivity for the KMS model.

Source p-Value prob4F

Model 0.0328 SignificantResidualLack of fit 0.1693 Not significantPure error

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266 263

model. Thus, statistically the model is sufficiently accurate in termof the prediction for yield and productivity within theexperimental range.

3.7. Optimisation of bioreactor's performances

3.7.1. Problem formulationOur objective is to find the optimum AR and SS by maximizing

the yield and productivity using the developed models, i.e. SDB,KH and KMS models. The optimisation problem is formulated asfollows:

P4 : maxXmin≤X≤XmaxðϕTϕÞ ð31Þ

Subject to: the model (either SDB, KH or KMS model).To solve this optimisation problem, we can use a nonlinear

programming technique, but in this study, we employ the RSMtechnique to find the optimum AR and SS.

3.7.2. Optimisation resultsAfter applying the RSM technique, a number of 3D surface plots

were generated for finding the optimum AR and SS using the SDBmodel as shown in Fig. 4. From Fig. 4, we observe that there is asignificant (quadratic) effect of AR and SS on the response surface.Within the experimental range, the optimum values of AR and SSmaximizing yield and productivity are 1.47 LPM and 242 rpm,

Design-Expert® Software

YieldDesign points above predicted valueDesign points below predicted value24.04

14.788

X1 = A: ARX2 = B: SS

200.00225.00

250.00

14

16.75

19.5

22.25

25

Yie

ld

B: SS

Design-Expert® Software

ProductivityDesign points above predicted valueDesign points below predicted value0.23

0.099

X1 = A: ARX2 = AB: SS

200.00225.00

250.00

0.09

0.125

0.16

0.195

0.23

Pro

duct

ivity

B: SS

Fig. 4. Response surface pl

respectively. The maximum values of yield and productivity are24.5% and 0.2 g/L h, respectively. This value of yield corresponds to97.8% of the maximum theoretical value for yield of ethanol. Ingeneral, the response of yield increases as the SS increases from150 rpm to its peak value at 242 rpm. Additionally, the yield showsa significant increase with the increase in AR. Overall, the SDBmodel demonstrates a reasonable prediction of the impacts of ARand SS on the values of yield and productivity since the ANOVAresults for SDB model, i.e. Tables 3 and 4 show the significance ofthis model on yield and productivity.

For the KH model, the effects of AR and SS on the yield andproductivity are shown in Fig. 5. The surface responses show thatboth yield and productivity are significantly affected by AR and SS.Also, it can be seen that the yield and productivity increase withthe increase of AR and SS. Thus, this suggests that the KH modelwas able to capture the effect of both AR and SS on the yield andproductivity. Just like the SDB model, the KH model is able topredict the impacts of AR and SS on the yield and productivityreasonably well.

The optimum values of AR and SS were obtained using the KHmodel as 1.43 LPM and 250 rpm, respectively. These optimumvalues lead to the maximum yield of 21.150% and maximumproductivity of 0.150 g/L h. Therefore, the maximum yield corre-sponds to 96.6% of the maximum theoretical value for yield. Also,we note that the predicted maximum yield and productivity by theKH model are comparable with that of the SDB model.

1.001.13

1.251.38

1.50

150.00175.00 A: AR

1.001.13

1.251.38

1.50

150.00175.00 A: AR

ots for the SDB model.

Design-Expert® Software

Yield21.099

12.006

X1 = A: ARX2 = B: SS

1.001.13

1.251.38

1.50

150.00 175.00

200.00 225.00

250.00

12

14.325

16.65

18.975

21.3

Yie

ld

A: ARB: SS

Design-Expert® Software

Productivity0.144

0.083

X1 = A: ARX2 = B: SS

1.001.13

1.251.38

1.50

150.00 175.00

200.00 225.00

250.00

0.083

0.099

0.115

0.131

0.147

Pro

duct

ivity

A: ARB: SS

Fig. 5. Response surface plots for the KH model.

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266264

Fig. 6 shows the response surface plots for the KMS model. Likeother models, the KMS model shows that the yield and produc-tivity increase with the increase of AR and SS. Within theexperimental range, the optimum values of AR and SS are1.45 LPM and 240 rpm, respectively. These values correspond tothe maximum yield of 24.128% and maximum productivity of0.207 g/L h. It is interesting to note, though, that the responsesurface plots generated based on the KMS model are almostsimilar to those generated by the SDB model. In contrast, theresponse surface plots of the KMS model are quite different fromthose of the KH model. This is due to the assumption of well-mixing behaviour in the macro bioreactor model used in theoriginal KH model.

3.7.3. Experimental validationIn this section, we present the experimental validation of the

optimum values of AR and SS found using different models.Tables 8 and 9 show the comparison of errors in the predictionsof maximum yield and productivity using different models. Theerrors were calculated as

Error Yield¼ 100%� jYieldmax−Yieldij=Yieldmax ð32Þ

Error Prod¼ 100%� jProdmax−Prodij=Prodmax ð33Þwhere the subscript i refers to the i-model, i.e. i∈ SDB;KH;KMSð Þ:

Note that the KMS model exhibits the best prediction ofmaximum experimental yield and productivity (lowest prediction

error). Despite its simplicity, the SDB model predictions arerelatively good and better than the KH model predictions. Thismeans that by including the CFD model in the macro bioreactormodel, the effect of mixing arising from the AR and SS canreasonably be captured by the KMS model — provide the mostaccurate prediction of yield and productivity. Meanwhile, the KHmodel is only capable of taking into account the effect of AR and SSwithin the context of well-mixing condition inside the bioreactor.The fact that KH model resulted in the largest error in thepredictions of maximum experimental yield and productivitysuggested that there was a significant deviation from ideal mixinginside the bioreactor. Interestingly, despite this significant devia-tion from the ideal mixing condition, the SDB model, whichdirectly expressed the yield and productivity as a quadraticfunction of AR and SS, was shown to be capable of predictingthe maximum yield and productivity with a sufficient accuracy i.e.less than 5% error.

To demonstrate that the non-ideal mixing conditions areoccurring in the bioreactor, Fig. 7 shows an example of the mixingphenomena obtained from the CFD simulation under the opera-tion of AR 1.47 LPM and SS 242 rpm. It was observed that both ARand SS have significant impacts towards the turbulence andmixing mechanism in the bioreactor. Thus, this simulation isshowing the occurrence of non-ideal mixing behaviour, andexplains the ability of the bioreactor models to capture the non-ideally mixing behaviour of the bioreactor, which is essential for abetter prediction of yield and productivity. Further research will bedone in order to fully evaluate this approach. Current study only

Design-Expert® Software

YieldDesign points above predicted valueDesign points below predicted value24.1

15.9

X1 = A: ARX2 = B: SS

1.001.13

1.251.38

1.50

150.00 175.00

200.00 225.00

250.00

15.9

18.125

20.35

22.575

24.8

Yie

ldA: ARB: SS

Design-Expert® Software

ProductivityDesign points above predicted valueDesign points below predicted value0.25

0.11

X1 = A: ARX2 = B: SS

1.001.13

1.251.38

1.50

150.00175.00

200.00225.00

250.00

0.11

0.145

0.18

0.215

0.25

Pro

duct

ivity

A: ARB: SS

Fig. 6. Response surface plots for the KMS model.

Table 8Comparisons of model predictions and experimental results for yield.

i-Model Optimumaerationrate (LPM)

Optimumstirrerspeed(rpm)

Modelpredictedmaximumyield (%)

Experimentallyverified maximumyield (%) Yieldi

Erroryield(%)

SDB 1.47 242 24.495 23.720 3.46KH 1.43 250 21.150 20.950 14.73KMS 1.45 240 24.128 24.570ℓ 1.80

ℓ —The yield obtained by using the AR and SS values from the KMS model isadopted as the experimentally maximum value, with respect to which the error iscalculated.

Table 9Comparisons of model predictions and experimental results for productivity.

i-Model Optimumaerationrate (LPM)

Optimumstirrerspeed(rpm)

Maximumproductivity(g/L h)

Experimentallyverified maximumproductivity (g/L h)Prodi

Errorprod(%)

SDB 1.47 242 0.198 0.185 11.90KH 1.43 250 0.150 0.148 29.52KMS 1.45 240 0.207 0.210ℓ 1.43

ℓ —The productivity obtained by using the AR and SS values from the KMSmodel is adopted as the experimentally maximum value, with respect to which theerror is calculated.

Fig. 7. CFD simulation of the mixing mechanism inside the bioreactor at AR of1.47 LPM and SS of 242 rpm (front view).

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266 265

shows the significant impacts of both AR and SS in terms of thebioreactor models developed in order to capture the non-ideallymixed behaviour of the bioreactor.

E.W.-T. Liew et al. / Chemical Engineering Science 95 (2013) 257–266266

4. Conclusions

In this paper, we have proposed a new approach to thebioreactor modelling based on the expanded Herbert's kineticsconcept. The developed models were able to incorporate themixing mechanism by adopting two input variables, namely theaeration rate and stirrer speed. It was shown that the models couldbe used to optimize the bioreactor's performances. Furthermore, itwas found from this study that the incorporation of mixing CFDmodel into the KH model of microbial kinetics (i.e. KMS modellingapproach) could predict reasonably well (and optimize) the yieldand productivity by adjusting the aeration rate and stirrer speed. Itis important to note that using the conventional models ofmicrobial kinetics it is not possible to optimize the yield andproductivity using the AR and SS because of well-mixed assump-tion. As the results, the developed models could be used forstudying the effects of AR and SS on the rates of growth, substrateconsumption and product formation so that the AR and SS couldbe used as additional parameters to optimize the bioreactor.

Acknowledgements

The work described in this paper was supported by theeScienceFund project (No: 02-02-07-SF0001) under the Ministryof Science, Technology and Innovation (MOSTI).

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