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Part Dynamic ioprocessSimulation Examplesand the erkeleyMadonna SimulationLanguage

Biological Reaction Engineering Second Edition. I. J.Dunn,E .Heinzle,J.Ingham,J. E. PfenosilCopyright 2 3W I L EY - V C H Verlag Gm bH Co. KGaA, W einheimISBN: 3-527-30759-1

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8 SimulationExamplesofBiologicalReaction Processes Using BerkeleyMadonna

8.1 Introductory Examples8 1 1 Batch Fermentation BATFERM)System

The system isrepresented in Fig. 1, and the important variables are biologicaldry mass or cell concentration, X, substrate concentration, S, and productconcentration, P. The reactor volume V is well-mixed, and grow th is assumedto follow kinetics described by the Monod equation, based on one limitingsubstrate. Substrate consu mption is related to cell grow th by a constant y ieldfactor YX/S- Product formation is the result of both growth and non-growthassociated rates ofprodu ction, where either termmay be set tozero as required.The lag and decline phases of cell growthare not included in the model.

Figure 1. Stirred batch fermenterwith model variables.

Biological ReactionEngineering, Second Edition. I. J. Dun n, E. Heinzle, J. Ingham, J. E. PfenosilCopyright2003W I L E Y - V CHVerlag Gm bH& Co. KGaA,W einheimISBN: 3-527-30759-1

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194 8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Model

MassBalances: (Rate of accumulation) = (Rate of production)For cellsV T T = r x v

or dXFor substrate

dSVd F = rsor dSdF = rSFor product dPV-3T =or dPdF = r?Kinetics:

rx = f i Xwith the Monod relation, constant yield relation, and product formationkinetics:

tx/srP = (ki +k2|^) X

where ki is the non-growth associated coefficient, and k2 is the coefficientassociated with growth.If the number of equations is equal to the number of unkn ow ns, the m odel iscomplete and the solution can beobtained. Theeasiest way to demonstrate thisis via an information flow diagram, as shown below in Fig. 2.

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8.1 Introductory Examples 195xo

So

P O

BiomassBalance4_

SubstrateBalance4_ProductBalance

A4

rx

rs

rp

T *GrowthRate

f'xSubstrateRate

ProductRate

M

-^M,-_

MonodKineticsA

X

s

p

Figure 2. Information flow diagram of the batch fermenter model equations,It is seen inthatall the variables required for the solution of any one equationblock are obtained as the products of other blocks. The inf orm ation flowdiagram thus emphasizes the complex inter-relationship involved in even thisvery simple problem. Solution begins with the initial conditions XQ , S Q and P Qat time t=0. The specific growth rate|iis calculated, enabling rs, rx and rp tobeca lculated, andhencethe initial gradients dX/dt, dS/dt anddP/dt. At this timethe integration routine takes over to estimate revised values of X, S and P overthe first integration step length. The procedure is repeated for succeeding steplengths until the entire X, S and P concentration time profiles have beencalculated up to the required final time.

ProgramThe following Berkeley Madonna program solves the above fermentationproblem:{BATPERM}{Batch growth with product formation}{Constants}U M = 0. 3KS= 0 1K l = 0 . 0 3K2=0.08Y = 0 .8

;kg/m3;kgP/kgX h;kgP/kgX h;kg X/kg

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X 0 = 0 . 0 1 /Initial biomass inoculum, kg/m3S O= 10 ;Initial substrate cone., kg/m3P 0 = 0 ;Initi al product conc.,kg/m3{Initial Conditions}INIT X=XOINIT S=SOINIT P=POMass Balances}X = RX ;BIOMASS BALANCES = RS ;SUBSTRATE BALANCEP RP ;PRODUCT BALANCE{Kinetics}RX = U*X ;BIOMASS RATE EQUATION, kg/m3 hU =UM*S/ KS+ S) ;MONOD EQUATION, 1/hRS =- R X / Y ;SUBSTRATE RATE EQUATION, kg/m3 hRP= K1 +2*U)*X /PRODUCT RATE EQUATION, g/m3 hLimit S>=0.0The semicolon or curly brackets are used for comments.INIT specifies the initial conditions. XQ , S Q and P Q are used here for the initialconditions, or the values at time=0. The fo rm X' designates the time derivativeor d/dt(X) can be used. Most models are conveniently structured in terms ofmass balances and kinetics. Any result quantity on the left of the equal sign isstored for further calculations or for use in graphing. U sually concentrationversustimeis ofinterest,but rates versus concentrations make very useful plotsfor understanding thekinetics. The five integration methods require specifyingtime intervals, such as DT, DTMIN and DTMA X. This requires a bit ofexperience. Care must be taken to see that the same results are obtained by twodifferent methods or for atleast two differentDT values.As is seen in the Appendix, Berkeley Madonna provides many possibilities tochange the parameters and graph new runs. These include the following:changing parameters with the parameter window and making overlay plots;changing parameters with sliders; usingthe Batch R uns facility.

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8.1 Introductory Examples

Nomenc la tureSymbolsk ] and k2KSPrSVXYH

Product formation constantsSaturat ion constantProduct concentrationReaction rateSubstrate concentrationReactor volumeBiomass concentrationY ield coeff icientSpecific growth rate

1/ h and kg/kgk g / m 3m g / m 3k g / m 3 h and kg /m3 hk g / m 3m3kg /m 3kg /kg1/h

197

Indices

2mPSX

Refers to non-g row th association rateRefers to growth-association rateRefers to m ax im umRefers to productRefers to substrateRefers to biomass

Exercises1. Vary KS,Mmseparately and observe the effects in the graphs. It isuseful to zoom in on regions of importance by using the zoom toolin thetool bar.2. Vary the pro duc t kinetics constants ( K j and K2>, and observe th eeffects. Observe the P versus time curve when Sreaches zero.3. Plot therates versus th e concentrations.

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198 8Simulation ExamplesofBiological Reaction Processes Using Berkeley MadonnaResults

Theplotsof X, S and PversusT inFig. 3 show that when substrate is depleted,thegrowthstops, and the product continues to increase, but only linearly. Theresults of Fig.4 are obtained by varying the product formation rate constants,ki in three runs using a slider, whichisdefinedin the Parameter Menu.Run 1:1500stepsin 0seconds

.10

Figure 3.Plots of X, S and Pversus time during batch growthan dproduction.

Run 3:1500 stepsin 0.0333 seconds....... l '

0 5 1 0

* '. _

-S:2P:2 -S:3P:3

-rpvtf*EV15TIME

\ /*

X//\/s \, ~'-'7 '

.** *'..

20 25 3(

-109-876.5 cn-43210)

Figure 4.Plots of P and Sversus time created by varying theproductformationrate constant

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8.1 Introductory Examples 199

8.1.2 Chemostat Fermentation CHEMO)SystemA continuous fermenter, as shown in Fig. 1, is referred to as a chemostat. Atsteady state the specific growth rate becomes equal to the dilution rate, |a= D.Operation ispossible at flowrates (F ) which give dilution rates (D = F/V) belowthe maximum specific growth rate (|um). Washout of the organisms will occurwhen D > ( a . The start-up, steady state and washout phenomena can beinvestigated by dynamic simulation.

D,SF

Figure 1. Chemostat with model variables.

Model

S X

The program BATFERM may be easily modified to allow for chemostatoperation with sterile feed by modifying the mass balance relationships toinclude the inlet andexit flow terms. The corresponding equations arethen:Forcells dX .= -DX +rx

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200 8 Simulation Examplesof Biological ReactionProcessesUsingBerkeley Madonna

For substrate dS3f = D(SF- S) +rsForproduct dPdf = -DP + rpwhereD is thedilution rate and Sp the concentrationof the limiting substrate inthe feed.The same kinetic expressionsas inBATFERMwillbeapplied here.

ProgramNote the conditional statement for Dwhich allowsabatch startup.{CHEMO}Chemostat startup and steady state. St art up asbatch reactor until time=tstart}{Constants}U M = 0 . 3 ; 1/hKS =0.1 ;kg/m3Kl=0.03 ; kgP/kgX hK2=0.08 ; kgP/kgXY=0.8 ; kg X/kgSX0 =0.01 ; Initial biomass inoculum, kg/m3S 0 = 1 0 ; Initia l substrate cone., kg/m3P0=0 ; Initial product conc.,kg/m3SF=0 ; Feed cone. ,kg/m3Dl =0.25 ;Dilution rate,1/htstart = ; Start time for the feed(Initial Conditions}Init X=XOInit S=SOInit P=PO{Mass Balances}X = - D * X + R X ; BIO MASS BALANCE EQUATIONS =D * SF-S)+RS ; SUBSTRATE BALANCE EQUATIO NP = - D * P + R P ; PRODUCT B ALANCE EQUATI ON

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8.1 Introductory Examples 2 1

{Kinetics}RX = U*X ; BIOMASS RATE EQUATION, kg/m3 hU =UM*S/ KS+ S) ; MONOD EQUATION, 1/hRS=-RX/Y ;SUBSTRATE RATE EQUATION, kg/m3 hRP= K1 +K2*U)*X ;PRODUCT RATE EQUATION, kg/m3 h{Conditional equation for D}D=if time>=tstart then Dl else 0P r o d = D * X /Productivity for biomass, kg/m3 h

Nomenclature ymbolsDkiKSPrSXYL L1

and Dilution rateProduct formation constantsSaturation constantProduct concentrationReaction rateSubstrate concentrationBiomass concentrationY ield coefficientSpecific growth rateTime lag constant

1/h1/h and kg/kgkg/m3mg/m 3kg/m 3h andkg/m 3kg/m 3kg/m 3kg/kg1/h

ndicesF Refersto feedMO NO D Refers to Monod kineticsP Refers to productS Ref ers to substrateX Refers to biomass

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202 8 Simulat ion Examples of Biological Reaction Processes Using Berkeley Madonna

Exercises

1. Increase D interactively to obtain wash out.2. N ote the steady state values of X and S; calculate Y from these.3. ChangeSF.Doesthi s alter S atsteady state? W hy ?4. Calculate S at steady state from D. Verify by simulation.5. Change th e program to account forbiomass in the feed.6. Operate initially as a batch reactor with D = 0, and switch tochemostat operation with D

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8.1 Introductory Examples 203Run 3:4000 stepsin 0 0333 seconds

10

Figure 2. Startupsof the chemostat after initial batch growth for 3valuesofDl.

Run8:200000 steps in1.38 seconds2

Figure 3 Productivity in achemostat. Steady states areshownfor 20 runs using the ParameterPlot.

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204 8 SimulationExamples ofBiological Reaction ProcessesUsingBerkeley Madonna8 1 3 Fed atch Fermentation FEDBAT)SystemIn this case the model equations allow for the continuous feeding of sterilesubstrate, the absence of outflow from the fermenter and the increase involumeaccumulation oftotal mass)in the fermenter, schematically asshowninFig. 1.Simulation of fed batch fermenters can be used to demonstrate th e importantcharacteristics of quasi-steady state, linear growth, and use of alternative feedstrategies.

F,SF

VXsp

Figure 1 Fed batch fermenter with model variables.

ModelFor fedbatch operation, theequations become as follows:Total balance

dVdT = FForcells

For substrate

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8.1 Introductory Exam ples 205

For product

where F is the volumetric feed rate, Spis the feed concentration and V is thevolume of the fermenter contents at time t. Thus the mass quantities, VX, VS,and VP are calculated and are divided by the volume at each time interval toobtain the concentration terms required for the kinetic relationships. Thekinetics aretakento be the sameas in BATFERM.

ProgramThe IF statementin the program causes thecontinuous feedto start when timereaches tfeed,atw hich point batch operation stopsand the fedbatch starts.FEDBAT){Fermentation with bat ch start up}{Flow rate is initially zero and is turned on att i m e = t f e e d . }{ Constants}U M = 0 . 3KS= 0 . 1Kl =0.03K2 =0.08Y =0.8X0 = 0 .01S0 = 10P0 = 0SF = 10Pl-1.5t f e e d = 2 2 . 5

; 1/h; kg/m3; kgP/kgX h; kgP/kgX; kg X/kg S; Initial biomass inoculum, kg/m3; Initial substrate cone., kg/m3; Initial product conc.,kg/m3; Feed conc.,kg/m3; Feed flow rate, m3/h; Start time for the feed

{Initialinit V = linit VX=V*X Oi n i t V S =V * S Oinit VP=V*PO

Conditions}

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{Mass balances, kg/h}d / d t ( V ) = Fd / d t ( V X ) = R X * Vd / d t ( V S ) = F * S F +R S *Vd / d t ( V P ) = R P * V {kg/h}{Calculation of concentrations}X=VX/VS=VS/VP=VP/V{Kinetics}R X = U * XU = U M * S / ( K S + S )RS=-RX/YRP= K1+K2*U)*XD=F/V {nominal dilution rate, 1/h}{Turning the feed on at time = tfeed}F=if time>=tfeed then Fl else 0 {batch start up}

NomenclatureSymbolsD Dilution rate 1/hF Flow rate m3/hKS Saturation constant kg/m3ki,k2 Constants in product kinetics 1/h andkg/kgM Maintenance coefficient kg/kg hP Product concentration kg/m3r Reaction rate kg/m3hS Substrate concentration kg/m3X Biom ass concentration kg/m3V Reactor volume m3Y Y ield coefficient kg/ kg|i Specific grow th rate 1/hT Time delay constant h

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8.1 Introductory Examples 207

ndicesFPSX

Refers to feedRefers to productRefers to substrateRefers tobiomass

Exercises

ResultsOperation begins under initial batch conditions, and feeding of substrate isstarted at tfeed= 22.5 h. In Fig. 2, the break in the b atch grow th transient, assemi-batch feeding starts is very apparent, with th e transient continuing to anapparent quasi steady state operating condition. U nder these cond itions thebiomass concentration becomes constant, while the substrate concentration (notshown) isbelow the KSvalue and decreases very slowly. As seen in the zoomof Fig. 3, the values of D (= F/V ) also decrease since V increases due to theincoming feed, and D eventually becomes equal to p when S falls below K$.The total biomass is determined by the yield coefficient times the total amountof substrate that has been consumed, which is approximately equal to theamount in the reactor initially plus th e amount added during th e feedingperiod. During the quasi-steady state, the total biomass will increase linearlywith time if, as in this case, the feeding flow rate is constant. This is a linear

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208 8 Simulation Examples of Biological ReactionProcesses Using Berkeley Madonna

growth situation in whichthe growth rate is limited by the feeding rate. InFig. 3 the values of X, S, and P are plotted versus T for a switch from batch(F =0) to fed batch (F = 5) at time T = 20 h. The product produ ction ratedepends linearly onbiomass concentration, and thus even when ja becomes verylow,P will continue to increase linearly inmg/m3amounts.

TIME= 34.13 X= 12.3410-.- .

10 20 30 40 50 60 70 80 90 100

Figure 2. Transients during the fedbatch fermentation.

0.4.

0 . 3 5 -0.3-

0.25.3Q 0.2-C O 0 . 1 5 -

0.1-0 .05-

0 -

-~, I-J.T

Run1 :5000stepsin 0.1 seconds

27 28 29 30 31 32 33 34 35 36 37TIME

Figure 3. Zooming in on the quasi-steady state.

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8.2Batch Reactors 209

8 2 BatchReactors

8.2.1 Kinetics of Enzyme Action MM KINET )SystemThe intermediate enzyme-substrate complex is the basis for the simplest formofenzymatic catalysis (Fig. 1):

E S^ ES - E Pk2

Figure 1. Mechanistic model for enzymatic reaction.

ModelThe equations for substrate, enzyme-substrate complex and product in a batchreactor are:

- = k iES - k2ESdtdFS = k iE S - (k2+k3)ES

dtUsing the steady state approximation for the change of active complex,

dtthe Michaelis-Menten equation isobtained.

_dS _~ dt ~ KM +S

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210 8 Simulation Examples ofBiological Reaction Processes Using Berkeley Madonna

wherevma x=k3E0andKM=(k2+k3)/ki.

ProgramThe program w ith the detailed mechanism is on the CD-R OM .

Nomenclature ymbolsEESkKMPSVmax

Enzyme concentration mol/m3Enzyme-substrate complex concentration mol/m3Reaction rate constants variousMichaelis-Menten constant mol/m3Product concentration mol/m3Substrate concentration mol/m3Maximum velocity mol/m3h

ndices0123SM m

Refers to initial valuesRefers to reaction1Refers to reaction 2Refers to reaction 3Refers tosubstrateRefers toMichaelis-Menten

Exercises

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8.2 Batch Reactors 211

ResultsFigs.2 and 3give the results of the full model and the Michaelis-Mentensimplification, respectively

Run 1: 11 9 steps in0.0167 seconds

0.009.0 .008-0 . 0 0 7 -0.006 tn0.005-

LU 0 .004-0 .003 -0.002 0.001

Lx ~\ f'\:. fv

riLt ~ f\1*i* \.i fc*v % %-, * '..._

..]...8:1ES:1--P:1

.0.9

-0.80.70.6 a0.5 *to0.40.30.2 0.1.0

10 20 30 40 50TIME

70 80 90 100

Figure 2. Results from the full model

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Figure 1 Lineweaver-Burk plot to determine vmand

ModelThe model is that of a batch reactor with Michaelis-Menten kinetics.

dSdF = ~rs

ProgramTo m ake the Linew eaver-B urk plot, the inverse values of S and rs are calculatedin the programon the CD-ROM.

Nomenclatureym ols

K MrS

Michaelis-Menten constantReaction rateSubstrate concentration

kg/m3kg/m 3kg/m 3

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214 8 Simulation Examples ofBiological Reaction Processes Using Berkeley MadonnaSiV

Inverse substrate concentrationReaction velocity or rateInverse reaction velocity or rate

m3/kgkg/m 3 hm

3h/kg

ndices0mS

Refers to feedRefers tom aximumRefers tosubstrate

Exercises

ResultsThe results are show n in Fig. 2 (rates and concen trations versus time) for arange of Michaelis-Menten constants KM and in Fig.3 the correspondingLineweaver-Burk plots.

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Run 4:13710stepsin 0.133 seconds0.50.45

.0.40.35.0.3.0.25 0.2-0.150.10.05

140 160

Figure 2. Rate and concentration plots for KM= 0.2, 0.5, 1.0 and 2.0 (bottom to top curves).Ru n 4:13710steps in 0.133 seconds

Figure 3 Lineweaver-Burk plots for KM= 0.2, 0.5, 1.0 and 2.0 (bottom to top curves).

8.2.3 Oligosaccharide Production in EnzymaticLactose Hydrolysis OLIGO)SystemSome enzy me catalyzed reactions are very com plex. For this reason theirrigorous modelling leads to complex kinetic equations with a large number ofconstants. Such models are unwieldy and are usually not suitable for practical

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purposes. One approach to simplify them is to neglect formation of enzyme-substrate complexes altogether and to deal only with overall reactions of thereactants to products.An example of such a reaction is the enzymatic lactose hydrolysis, acomplex process involving amultitude of sequential reactions leading to highersaccharide (oligosaccharides) intermediates. The mechanistic model is rathercomplex even when only trisaccharides are considered (Fig. 1).

La+ E LaE ^ Ga+GI+ EGaE La E TrGaE+H2O E +Ga

Figure 1.Complex an d simplified models for the enzymatic hydrolysis of lactose, where thesymbols are La for lactose, Ga forgalactose, Gl forglucose, Tr fortrisaccharideand E forenzyme.

Neglecting the enzyme complexes, however, gives a simplified model (Fig. 2)requiring only three constants:

a.a

Lac l

K

-i- Ga

to Ga

K1K2

. f^ iT V3II

Tri

Figure 2. Simplified model for the enzymatic hydrolysis of lactose.

The simulation of this model is easy, and the constants can be adjusted toachieve good agreement with experimental data.

Model

This simple batch reactor model isequivalent to the Michaelis-Menten productinhibition model.

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dL a-gj-dGa

- K La - KILa Ga +K2Tr

= KjLa - KILa Ga +K2TrdTr = KILa Ga - K2Tr

Initial conditions: Lao=150mmol/m3,Gao= 0,Trg= 0Rangeof thekinetic constants: KI = 0.02 - 0.06 miir1, K I = 0.02 - 0.1

L/mmol min,K2= 1 - 50min 1.ProgramIt wasfound thatK2mustbe twoorders of magnitude greater than K I in orderto bring the simulation into agreement with the experimental data. Theprogram is on the CD-ROM.

Nomenclature ymbolsGaGl

K2L aTr

Galactose concentration mmol/LGlucose concentration mmol/LReaction rate constant (L a > Ga + Gl) 1/minReaction rate constant(La + Ga->Tri) L/(mmol min)Reaction rate constant (Tri ->La + Ga) 1/minLactose concentration mm ol/LTrisaccharide concen tration mm ol/L

ndices0 Refers to initial concentration

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Exercises

ResultsThe outputs inFigs.3 and 4 show th einfluenceofK I , KIand Laoon the sugarconcentration profiles.

100.90.80.706 0 .

. 5 0 .4030201 0

0

Run1:10000stepsin0.05secondsr100

20 100TIME

180 200

Figure 3. Sugar concentrations with Kr = 0.04, K { = 0.05, La0 = 100.

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Run1 :10000 stepsin0.15 seconds160

80 100 120 140 160 180 200

(80

Figure 4. Sugar concentrations with Kj= 0.06, KI = 0 .1 L ao= 160.

ReferencePrenosil, J. E., Stuker, E. and Bourne, J. R. (1987) Formation ofOligosaccharides during an Enzymatic Lactose Hydrolysis Process , Parts I andII : Biotechnol. Bioeng. 30, 1019-1031.

8 2 4 Structured Model for PHB Production PHB)System

Heinzle and Lafferty (1980) have presented a structured model to describe thebatch culture of Alcaligenes eutrophus under chemolithoautotrophic growthconditions,asdiscussed inCase C,Sec. 3.3.1. Growthand storage of PHB aredescribed as functions of limiting substrate S (NH4+), residual biomass R andproduct P (PHB) concentrations.

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220 8Simulation Examples of Biological ReactionProcesses Using Berkeley Madonna

Figure 1.Structured kinetic model for PHB synthesis.

ModelIn the model seen in Fig. 1 the whole cell dry mass (X ) consists of two mainparts, namely PHB (P) and residual biomass (R), where R is calculated as thedifference between the totalcell dry weight and the concentration of PHB (R =X - P). R can be considered as the catalytically active biomass, includingproteins and nucleic acids. With constant concentrations of the dissolved gases,two distinct phases can be recognized: grow th and storage. Du ring the grow thphase there is sufficient N H4+ to permit protein synthesis. W hen the limitingsubstrate NH4+ (S) is exhausted, the protein synthesis ceases, and theproduction rate of PHB is increased. Du ring the storage phase on ly PHB isproduced. The limiting substrate NH4+ (S) isessential to produce R and limitsits synthesis at low concentrations.For the batch process,

dRdF = rR = M R

where T R is the rateof synthesisof R and (jis the specific rate of synthesis of R,whereS (S/Ks,2)n

+S) + ^m,2+(S/KS,2)nwhere n is the empirical Hill coefficient (see Sec. 3.1.2), having a value of 4 inthis example.

This is based on the postulate that there are two different mech anisms for theassimilation of NH4+ inprocaryotes. This form ulationis not a mechanistic one,

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8.2Batch Reactors 221

since in reality the enzyme system, using energy to assimilate NH4+, isrepressed by high concentrations of NH4+.

For the substrate dS 1d F = rs = - Y R / S * *The rate of synthesis of P(rp) is assumed to be the sum of a growth associatedterm (rpj) and a biomass associated term (rp,2)and is given by,

dPdf = rp = rPj+rP,2whererPj = YP/R rRThe non-growth associated term of the synthesis of P(rp,2) is assumed to be afunction of the limiting substrate S, of the residual biomass R and of theproduct P. W hen the PHB content in the cells is high, the rate of synthesis of Pisdecreased, which can be formally described as an inhibition.

ProgramThe program is found on the CD-ROM.

Nomenclatureym olsK I Inhibition constant, for(NH^SC kg /m3KS Saturation constant kg/m3n Hill Coe fficientP Product concentration (PHB ) kg/m3R R esidual biomass concentration kg/m3rp Rateof synthesisof PHB kg/m3T R Rate of synthesisof R kg/m3rs Rateof substrate uptake kg/(m3h)

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XYP/RYR/S

Limiting substrate concentration kg/m3NHj as(NH4)2S04Biomass concentration kg/m3Y ield coefficient kg/ kgY ield coefficient, kg/ kgSpecific rateof synthesisof R (rR/R) 1/hSpecific rate of synthesisof P (rp/P) 1/h

ndices

12mRefers toreaction 1Refers toreaction 2Refers to maximum

Exercises

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Resul tsRun 1:416stepsin0.0167 seconds

4-13.5-

3-2.5-

of 2-1.5-

1-0.5-

0-

~ ~*"f -*/ / ISi$%-. T M . T.-'VT

' U--b* ' -'' J '*y '/ f/ ' . .^ V* _ j _ _ * %

-161412-10-8 a.-64-2_ n

0 5 10 15 20 25 30 35 40TIME

Figure 2 * Profiles of residual biomass concentration R, substrate S andproduct P in the batchfermentation.Run4: 416stepsin 0.0167 seconds

35-

30-25-

20 -a15-

1 0 -5

0.

...' 'V ^ -

'v\ / .._.3:3(2,3)\ ~-P:3(2.3)1 / P:4(5)-*. \ /

"""-, \ f .* -'\ \i'"" "' /^^ ";:

54.5-43.53-2.5 (/)-2-1.5-1

-0.5-n0 5 10 15 20 25 30 35 40

TIME

Figure 3. PHB formation at two different initial substrate concentrations.

ReferencesHeinzle, E., and Lafferty, R. M. (1980) Continuous Mass SpectrometricMeasurement of Dissolved H2, O2, and C C > 2 during Chemolitho-autotrophic

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Growth of Alcaligenes eutrophus strain H16. Eur. J. Appl. M icrobiol.BiotechnoL 11, 8.1

8 3 Fed Batch Reactors8 3 1 Variable Volume Fermentation VARVOL andV A R V O L D )SystemSemi-continuous or fed batch cultivation of micro-organisms is common in thefermentation industries. The fed batch fermenter m ode is show n inFig.1 andwas also presented in the example FEDB AT. In this procedu re a substrate feedstream is added continuously to the reactor. After the tank is full or thebiomass concentration is too high, the medium can be partially emptied, andthefilling process repeated. Since the variables, volum e, substrate and biom assconcentration change with time, simulation techniques are useful in analyzingthis operation. This example demonstrates the use of dimensionless equations.

Figure 1.Filling and emp tying sequences in a fed batch fermenter.

ModelThe balances are as follows:Volume, dv

dT = FO

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